UNIVERSITY  OF  CALIFORNIA 
AT   LOS  ANGELES 


I  below. 


SOU 


1U>S  ANG£i_L.S.  CALIF.' 


AN  INTRODUCTION  TO  THE  USE 
OF  GENERALIZED  COORDINATES 
IN  MECHANICS  AND  PHYSICS 


BY 


WILLIAM  ELWOOD  BYEELY 

PERKIXS  PKOFESSOB  OF  MATHEMATICS  EMERITUS 
IX  HARVARD  UNIVERSITY 


44922 


GINN  AND  COMPANY 

BOSTON     •     NEW    YORK     •     CHICAGO     •     LONDON 
ATLANTA     •     DALLAS     •     COLUMBUS     •     SAN   FRANCISCO 


COPYRIGHT,  1916,  BY 
WILLIAM  EL  WOOD  BYERLY 


ALL  RIGHTS  RESERVED 
11G.6 


Cfre   gtfttnacutn    ffrcss 

GINN  AND  COMPANY  •  PRO- 
PRIETORS •  BOSTON  •  U.S.A. 


Engineering  & 

Mathematical 

Sciences 

Library 


QA 

\ 


PKEFACE 

This  book  was  undertaken  at  the  suggestion  of  my  lamented 
colleague  Professor  Benjamin  Osgood  Peirce,  and  with  the  promise 
of  his  collaboration.  His  untimely  death  deprived  me  of  his  invalu- 
able assistance  while  the  second  chapter  of  the  work  was  still 
unfinished,  and  I  have  been  obliged  to  complete  my  task  without 
the  aid  of  his  remarkably  wide  and  accurate  knowledge  of  Mathe- 
matical Physics. 

The  books  to  which  I  am  most  indebted  in  preparing  this  treatise 
are  Thomson  and  Tait's  "Treatise  on  Natural  Philosophy,"  Watson 
and  Burbury's  "  Generalized  Coordinates,"  Clerk  Maxwell's  "  Elec- 
tricity and  Magnetism,"  E.  J.  Routh's  "  Dynamics  of  a  Rigid 
Body,"  A.  G.  Webster's  "  Dynamics,"  and  E.  B.  Wilson's  "  Advanced 
Calculus." 

For  their  kindness  in  reading  and  criticizing  my  manuscript  I  am 
indebted  to  my  friends  Professor  Arthur  Gordon  Webster,  Professor 
Percy  Bridgman,  and  Professor  Harvey  Newton  Davis. 

W.  E.  BYERLY 


iii 


CONTENTS 

CHAPTER  I 

INTRODUCTION 1~37 

ART.  1.  Coordinates  of  a  Point.  Number  of  degrees  of  freedom.  — 
ART.  2.  Dynamics  of  a  Particle.  Free  Motion.  Differential  equa- 
tions of  motion  in  rectangular  coordinates.  Definition  of  effective 
forces  on  a  particle.  Differential  equations  of  motion  in  any  sys- 
tem of  coordinates  obtained  from  the  fact  that  in  any  assumed 
infinitesimal  displacement  of  the  particle  the  work  done  by  the 
effective  forces  is  equal  to  the  work  done  by  the  impressed  forces. 
—  ART.  3.  Illustrative  examples. — ART.  4.  Dynamics  of  a  Particle. 
Constrained  Motion.  —  ART.  5.  Illustrative  example  in  constrained 
motion.  Examples.  —  ART.  6  (a).  The  tractrix  problem.  (6).  Par- 
ticle in  a  rotating  horizontal  tube.  The  relation  between  the  rec- 
tangular coordinates  and  the  generalized  coordinates  may  contain 
the  time  explicitly.  Examples.  —  ART.  7.  A  System  of  Particles. 
Effective  forces  on  the  system.  Kinetic  energy  of  the  system. 
Coordinates  of  the  system.  Number  of  degrees  of  freedom.  The 
geometrical  equations.  Equations  of  motion.  — ART.  8.  A  Sys- 
tem of  Particles.  Illustrative  Examples.  Examples. — ART.  9.  Rigid 
Bodies.  Two-dimensional  Motion.  Formulas  of  Art.  7  hold  good. 
Illustrative  Examples.  Example.  —  ART.  10.  Rigid  Bodies.  Three- 
dimensional  Motion,  (a).  Sphere  rolling  on  a' rough  horizontal  plane. 
(b).  The  billiard  ball.  Example,  (c).  The  gyroscope,  (d).  Euler's 
equations.  —  ART.  11.  Discussion  of  the  importance  of  skillful 
choice  of  coordinates.  Illustrative  examples.  —  ART.  12.  Nomen- 
clature. Generalized  coordinates.  Generalized  velocities.  Gen- 
eralized momenta.  Lagrangian-  expression  for  the  kinetic  energy. 
Lagrangian  equations  of  motion.  Generalized  force.  —  ART.  13. 
Summary  of  Chapter  I. 

CHAPTER  II 

THE    HAMILTONIAN    EQUATIONS.    ROUTH'S    MODIFIED    LA- 

GRANGIAX  EXPRESSION.  IGNORATION  OF  COORDINATES  .  38-61 
ART.  14.  IlamiUonian  Expression  for  the  Kinetic  Energy  defined. 
Hamiltonian  equations  of  motion  deduced  from  the  Lagrangian 
Equations.  —  ART.  15.  Illustrative  examples  of  the  employment  of 
Hamiltonian  equations.  —  ART.  16.  Discussion  of  problems  solved 
in  Article  15.  Ignoring  coordinates.  Cyclic  coordinates.  Ignorable 


vi  CONTENTS 

coordinates. — ART.  17.  South' s  Modified  Form  of  the  Lagrangian 
Expression  for  the  Kinetic  Energy  of  a  moving  system  enables  us 
to  write  Hamiltonian  equations  for  some  of  the  coordinates  and 
Lagrangian  equations  for  the  rest.  —  ART.  18.  From  the  Lagran- 
gian expression  modified  for  all  the  coordinates  we  can  get  valid 
Hamiltonian  equations  even  when  the  geometrical  equations  con- 
tain the  time  explicitly.  — ART.  19.  Illustrative  example  of  the  use 
of  Hamiltonian  equations  in  a  problem  where  the  geometrical  equa- 
tions contain  the  time.  Examples.  —  ART.  20.  Illustrative  examples 
of  the  employment  of  the  modified  form.  Examples.  —  ART.  21. 
Discussion  of  the  problems  solved  in  Article  20.  Ignoration  of  co- 
ordinates. —  ART.  22.  Illustrative  example  of  ignoring  coordinates. 
Example.  —  ART.  23.  A  case  where  the  contribution  of  ignored 
coordinates  to  the  kinetic  energy  is  ignorable.  Example.  —  ART. 
24.  Important  case  where  the  contribution  of  ignorable  coordinates 
to  the  kinetic  energy  is  zero.  Example  of  complete  ignoring  of 
coordinates  in  a  problem  in  hydromechanics.  —  ART.  25.  Summary 
of  Chapter  II. 

CHAPTER  III 

IMPULSIVE  FORCES 62-80 

ART.  26.  Virtual  Moments.  Definition  of  virtual  moment  of  a  force 
or  of  a  set  of  forces.  — ART.  27.  Equations  of  motion  for  a  particle 
under  impulsive  forces  (rectangular  coordinates).  Effective  impul- 
sive forces.  Virtual  moment  of  effective  forces  is  equal  to  virtual 
moment  of  actual  forces.  Lagrangian  equation  for  impulsive  forces. 
Definition  of  impulse. — ART.  28.  Illustrative  Examples  of  use  of 
Lagrangian  equations  where  forces  are  impulsive.  —  ARTS.  29-31. 
General  Theorems  on  impulsive  forces  :  ART.  29,  General  Theorems. 
"Work  done  by  impulsive  forces.  ART.  30,  Thomson's  Theorem. 
ART.  31,  Bertrand's  Theorem.  Gauss's  Principle  of  Least  Con- 
straint. —  ART.  32.  Illustrative  examples  in  use  of  Thomson's 
Theorem.  —  ART.  33.  In  using  Thomson's  Theorem  the  kinetic 
energy  may  be  expressed  in  any  convenient  way.  Example.  — 
ART.  34.  Thomson's  Theorem  may  be  made  to  solve  problems  in 
motion  under  impulsive  forces  when  the  system  does  not  start 
from  rest.  Example.  —  ARTS.  35-36.  A  Problem  in  Fluid  Motion. 
—  ART.  37.  Summary  of  Chapter  III. 

CHAPTER  IV 

CONSERVATIVE  FORCES 81-97 

ART.  38.  Definition  of  force  function  and  of  conservative  forces. 
The  work  done  by  conservative  forces  as  a  system  passes  from  one 
configuration  to  another  is  the  difference  in  the  values  of  the  force 


CONTENTS  vii 

function  in  the  two  configurations  and  is  independent  of  the 
paths  by  which  the  particles  have  moved  from  the  first  configura- 
tion to  the  second.  Definition  of  potential  energy.  —  ART.  39. 
The  Lagrangian  and  the  Hamiltonian  Functions.  —  Canonical 
forms  of  the  Lagrangian  and  of  the  Hamiltonian  equations  of 
motion.  The  modified  Lagrangian  function  <£.  —  ART.  40. 
Forms  of  L,  H,  and  <£  compared.  Total  energy  E  of  a  system 
moving  under  conservative  forces.  —  ART.  41.  When  i,  H,  or  * 
is  given,  the  equations  of  motion  follow  at  once.  When  L  is 
given,  the  kinetic  energy  and  the  potential  energy  can  be  dis- 
tinguished by  inspection.  When  H  or  <1>  is  given,  the  potential 
energy  can  be  distinguished  by  inspection  unless  coordinates 
have  been  ignored,  in  which  case  it  may  be  impossible  to  sep- 
arate the  terms  representing  potential  energy  from  the  terms 
contributed  by  kinetic  energy.  Illustrative  example. — ART.  42. 
Conservation  of  Energy  a  corollary  of  the  Hamiltonian  canonical 
equations. — ART.  43.  Hamilton's  Principle  deduced  from  the 
Lagrangian  equations.  —  ART.  44.  The  Principle  of  Least  Action 
deduced  from  the  Lagrangian  equations.  —  ART.  45.  Brief  dis-. 
cussion  of  the  principles  established  in  Articles  43-44.  — ART.  46. 
Another  definition  of  action. — ART.  47.  Equations  of  motion 
obtained  in  the  projectile  problem  (a)  directly  from  Hamilton's 
principle,  (b)  directly  from  the  principle  of  least  action.  — 
ART.  48.  Application  of  principle  of  least  action  to  a  couple 
of  important  problems.  —  ART.  49.  Varying  Action.  Hamilton's 
characteristic  function  and  principal  function. 

CHAPTER  V 

APPLICATION  TO  PHYSICS       .     . 98-108 

ARTS.  50-51.  Concealed  Bodies.  Illustrative  example. — ART.  52. 
Problems  in  Physics.  Coordinates  are  needed  to  fix  the  elec- 
trical or  magnetic  state  as  well  as  the  geometrical  configuration 
of  the  system.  —  ART.  53.  Problem  in  electrical  induction. — 
ART.  54.  Induced  Currents. 

APPENDIX  A.    SYLLABUS.  DYNAMICS  OF  A  EIGID  BODY  109-113 
APPENDIX  B.    THE  CALCULUS  OF  VARIATIONS  .  114-118 


GENERALIZED  COORDINATES 

CHAPTER  I 
INTRODUCTION 

1.  Coordinates  of  a  Point.  The  position  of  a  moving  particle 
may  be  given  at  any  time  by  giving  its  rectangular  coordinates 
z,  «/,  z  referred  to  a  set  of  rectangular  axes  fixed  in  space.  It 
may  be  given  equally  well  by  giving  the  values  of  any  three 
specified  functions  of  #,  y,  and  2,  if  from  the  values  in  question 
the  corresponding  values  of  #,  y,  and  z  may  be  obtained  uniquely. 
These  functions  may  be  used  as  coordinates  of  the  point,  and 
the  values  of  #,  y,  and  z  expressed  explicitly  in  terms  of  them 
serve  as  formulas  for  transformation  from  the  rectangular  sys- 
tem to  the  new  system.  Familiar  examples  are  polar  coordi- 
nates in  a  plane,  and  cylindrical  and  spherical  coordinates  in 
space,  the  formulas  for  transformation  of  coordinates,  being 
respectively 

x  =  r  cos  <f>,  1  x  =  r  cos  0, 

,£    -    ^.    QQg   (f)       I 

H    (1)      w  =  rsin<£,  \    (2)       y  =  r  sin  0  cos<£,  }•    (3) 
y  =  r  sin  <f>, 


z  =  z,  J  z  =  r  sin  6  sin 


It  is  clear  that  the  number  of  possible  systems  of  coordinates 
is  unlimited.  It  is  also  clear  that  if  the  point  is  unrestricted  in 
its  motion,  three  coordinates  are  required  to  determine  it.  If  it 
is  restricted  to  moving  in  a  plane,  since  that  plane  may  be  taken 
as  one  of  the  rectangular  coordinate  planes,  two  coordinates  are 
required. 

The  number  of  independent  coordinates  required  to  fix  the 
position  of  a  particle  moving  under  any  given  conditions  is 
called  the  number  of  degrees  of  freedom  of  the  particle,  and  is 


2  INTEODUCTION  [AKT.  2 

equal  to  the  number  of  independent  conditions  required  to  fix 
the  point. 

Obviously  these  coordinates  must  be  numerous  enough  to  fix 
the  position  without  ambiguity  and  not  so  numerous  as  to  render 
it  impossible  to  change  any  one  at  pleasure  without  changing 
any  of  the  others  and  without  violating  the  restrictions  of  the 
problem. 

2.  Dynamics  of  a  Particle.  Free  Motion.  The  differential 
equations  for  the  motion  of  a  particle  under  any  forces  when 
we  use  rectangular  coordinates  are  known  to  be 

mx  —  X,  ~\ 

my  =  F,   \  (1)* 

mz  =  Z.  J 

X,  F,  and  Z,  the  components  of  the  actual  forces  on  the  particle 
resolved  parallel  to  the  fixed  rectangular  axes,  or  rather  their 
equivalents  mx,  my,  mz,  are  called  the  effective  forces  on  the  parti- 
cle. They  are  of  course  a  set  of  forces  mechanically  equivalent 
to  the  actual  forces  acting  on  the  particle. 

The  equations  of  motion  of  the  particle  in  terms  of  any  other 
system  of  coordinates  are  easily  obtained. 

Let  q^  q2,  qs,  be  the  coordinates  in  question.  The  appropriate 
formulas  for  transformation  of  coordinates  express  x,  y,  and  z  in 
terms  of  q^  q2,  and  qs. 


For  the  component  velocity  x  we  have 

dx  .        dx  .        ex  . 


x  = 


and  x,  y,  and  z  are  explicit  functions  of  q^  q2,  q3,  q^  q2,  and  qs, 
linear  and  homogeneous  in  terms  of  q^  q2,  and  qa. 

*  For  time  derivatives  we  shall  use  the  Newtonian  fluxion  notation,  so  that 


dx  x 

we  shall  write  x  for  —  ,  x  for  -- 
dt  dt2 


CHAP.  I]  FREE  MOTION  OF  A  PAETICLE  3 

We  may  note  in  passing  that  it  follows  from  this  fact  that 
a?,  y2,   and  z2  are  homogeneous  quadratic  functions  of  q^  q2, 

fix       dx 
Obviously  —  =  —  5  (2) 


and  since 

d  dx\    d2x  . 

c2x     .          d*x     . 

/> 

dt  dq       dq^    * 

^o^l     2         ^^a^?]     ' 

and       ' 
Mf 

dx      d*x  . 

+                    A      1                      A 

(3) 

o         •        Q     2  ?1 

d 

dt 

o       5       2  2  ""     o       o       ?8' 

^X           Ci 

^<y,     ^<7, 

Let  us  find  now  an  expression  for  the  work  S?iJFdone  by  the 
effective  forces  when  the  coordinate  q1  is  changed  by  an  infini- 
tesimal amount  8q1  without  changing  q2  or  qa.  If  Sx,  8y,  and 

8z  are  the  changes  thus  produced  in  x,  y,  and  z,  obviously 

- 


Bg  W  —  m  [x&x  +  ij$y  +  2 

if  expressed  in  rectangular  coordinates.    We  need,  however,  to 
express  BqW  m  terms  of  our  coordinates  q^  q^  and  q3. 


..dx       d     .  dx\       .  d  dx 

Now  x—-  =  —lx—]  —  x  —  -— 

cq^      dt\    cqj         dtoq^ 

but  by  (2)  and  (3) 

3x       dx  ,      d  dx      ''"dx 

~ 


..dx       d  /  .  dx\      .dx       d    d  /x 
Hence     x—  -  =  —l  x-—  I—  x—  =  — 


i      dt\    dq 
and  therefore  ^r-.-afe  (4) 


lYfl 

where  T  =  •?-  [i?  +  if  +  z* 

2. 
and  is  the  kinetic  energy  of  the  particle. 


4  INTRODUCTION  [ART.  3 

To  get  our  differential  equation  we  have  only  to  write  the 
second  member  of  (4)  equal  to  the  work  done  by  the  actual 
forces  when  q:  is  changed  by  8q^ 

If  we  represent  the  work  in  question  by  Qfiqv,  our  equation  is 

ItZ  —  .Q,  (5) 

at  cq^      cq1 

and  of  course  we  get  such  an  equation  for  every  coordinate. 

It  must  be  noted  that  usually  equation  (5)  will  contain  q2 
and  qs  and  their  time  derivatives  as  well  as  q^  and  therefore  can- 
not be  solved  without  the  aid  of  the  other  equations  of  the  set. 

In  any  concrete  problem,  T  must  be  expressed  in  terms  of  q^ 
qz,  qs,  and  their  time  derivatives  before  we  can  form  the  expres- 
sion for  the  work  done  by  the  effective  forces.  Qfiq?  Qfiq^ 
Qs&q3,  the  work  done  by  the  actual  forces,  must  be  obtained 
from  direct  examination  of  the  problem. 

3.  (a)  As  an  example  let  us  get  the  equations  hi  polar  coor- 
dinates for  motion  in  a  plane. 

'Here  #  =  rcos<£,         y  =  r  sin  <f>. 


and 


?\T 

V  -*-  _L2 

—  =  mrq> . 
dr 

if  72  is  the  impressed  force  resolved  along  the  radius  vector. 

cT 


CHAP.  I]  ILLUSTRATIVE  EXAMPLES  5 

if  <I>  is  the  impressed  force  resolved  perpendicular  to  the  radius 
vector. 

In  more  familiar  form 

72~     '       //7JA2n 

=  tf, 


r  dt\     dt 
In  cylindrical  coordinates  where    <- 

x  =  r  cos  </>,         y  —  r  sin  <f>,         2  =  2, 


cT 
-— 

cr 

°T 
—  - 
dr 

dT 


8T 

—  =  mz. 
cz 

BrW  =  m  [r  —  r^2]  8r  =  R8r, 


or  m 


~'m~j~ 
V  =  mzSz  =  ZSz ; 

72™ 


m  d  I  „  dd> 

—   r2  -2 

r  dt\     dt 


INTRODUCTION  [ART.  4 

In  spherical  coordinates  where 
x  =  r  cos  0,         y  =  r  sin  6  cos  </>,         z  =  r  sin  6  sin  <£, 


ar=    . 

3r  ~ 

C-^-  =  mr  [<?2  +  sin2  6><621 


9        •  /I  /I    |0 

—  -  =  mir  sin  0  cos  0</>  , 

CU 


—  r  =  mr  sin  v<>. 
3<j> 

=  »i  [r  -  r  (02  +  sin2 

=  m  T^-  (r2^)  -  r2  sin  (9  cos  d       80 

=  TW        r2  sin2  0<>   S<>  =  <l>r  sin 


,      /) 

or  w  ^  —  —  r     -—    +  sm20  -f-]     L  = 


rldt\    dt/  \dt 

m      d  I  o   .  o  ^d 
—  -  —   r2  sui2  (9  -"  = 


r  sin 


4.  Dynamics  of  a  Particle.  Constrained  Motion.  If  the  particle 
is  constrained  to  move  on  some  given  surface,  any  two  inde- 
pendent specified  functions  of  its  rectangular  coordinates  x,  y,  z, 
may  be  taken  as  its  coordinates  q1  and  <?2,  provided  that  by  the 
equation  of  the  given  surface  in  rectangular  coordinates  and 
the  equations  formed  by  writing  q1  and  q^  equal  to  their  values 


CHAP.  I]     CONSTRAINED  MOTION  OF  A  PARTICLE  7 

in  terms  of  x,  y,  and  z  the  last-named  coordinates  may  be 
uniquely  obtained  as  explicit  functions  of  q1  and  q2.  For  when 
this  is  done,  the  reasoning  of  Art.  2  will  hold  good. 

If  the  particle  is  constrained  to  move  in  a  given  path,  any 
specified  function  of  x,  y,  and  z  may  be  taken  as  its  coordinate 
q^  provided  that  by  the  two  rectangular  equations  of  its  path 
and  the  equation  formed  by  writing  ql  equal  to  its  value  in 
trrnis  of  a*,  y,  and  z  the  last-named  coordinates  may  be  uniquely 
obtained  as  explicit  functions  of  qr  For  when  this  is  done,  the 
reasoning  of  Art.  2  will  hold  good. 

5.  (a)  For  example,  let  a  particle  of  mass  m,  constrained  to 
move  on  a  smooth  horizontal  circle  of  radius  a,  be  given  an 
initial  velocity  F,  and  let  it  be  resisted  by  the  air  with  a  force 
proportional  to  the  square  of  its  velocity. 

Here  we  have  one  degree  of  freedom.  Let  us  take  as  our 
coordinate  ql  the  angle  6  which  the  particle  has  described  about 
the  center  of  its  path  in  the  time  t 


_  m 
=  "2 

ar 

=  ma?V. 

CO 

Our  differential  equation  is 

^  / 


and  we  have  — r  =  ma20. 


which  reduces  to  0  +  -a£=0, 

m 

d6      Tc    A2     A 

or  —  +  —  ad2  =  0. 

at      m 

Separating  the  variables, 

dO      k 

-  +  —  adt  =  0. 
0*      m 

1       k  a 

integrating,         1 —  at  =  C  = • 

$      m 

-<±  --C 
ir 


INTRODUCTION  [ART.  5 

1  _  ma  +  k  Vat 
Q  mV 

6  mV 


6>  =       log  [>  +  m]  +  (7, 
fed 

a      m  ,      [.,      &Ff| 

0  =  7-log  1  +  —  ;  (1) 

ka        L         w  J 

and  the  problem  of  the  motion  is  completely  solved. 

(6)  If,  however,  we  are  interested  in  R,  the  pressure  of  the 
constraining  curve,  we  must  proceed  somewhat  differently.  We 
have  only  to  replace  the  constraint  by  a  force  R  directed  toward 
the  center  of  the  path.  There  are  now  two  degrees  of  freedom, 
and  we  shall  take  6  and  the  radius  vector  r  as  our  coordinates 
and  form  two  differential  equations  of  motion. 


cT  .2* 

—  =  mr20, 


dT 


— 


m(r-  rff2)  8r  =  - 

To  these  we  may  add 

r  =  a. 

Whence  6  +  —  8*  =  0,  as  before,  (3) 

m 


and  R  =  ma.  (4) 


CHAP.  I]     CONSTRAINED  MOTION  OF  A  PARTICLE  9 

(<?)  Let  us  now  suppose  that  the  constraining  circle  is  rough. 
Here,  since  the  friction  is  /-t  (the  coefficient  of  friction)  multi- 
plied by  the  normal  pressure,  B  will  be  needed,  and  we  must 
replace  the  constraint  by  R  as  before. 

We  have  now 

m  4  (W)  80  =  -  krs0*S0  - 
at 


and  r  =  a. 

Whence  ""  R  =  ma0\  as  before. 


m          ma 


Replacing  —  in  Art.  5,  fa),  (1),  by 1-  u, 

m  m 

we  have  0  =  - log   l  +  (—  +  /*)—  •  (1) 

ka  °L        \m         '  a\ 

m 

EXAMPLES 

1.  Obtain    the    familiar    equation    — --f-srn0  =  0    for   the 

,   ,  dr      a 

simple  pendulum. 

"-  2.  Find  the  tension  of  the  string  in  the  simple  pendulum. 

f  /<^Y1 

Ans.  R  =  m\  q  cos  u  +  a  -       I • 

L^  \dt/\ 

3.  Obtain  the  equations  of  the  spherical  pendulum  in  terms 
of  the  spherical  coordinates  0  and  <f>. 

Ans.  0  -  sin  0  cos  0<j>2  +  9-  sin  0  =  0.         sin2  00  =  C. 
a 

6.  (a)  The  constraint  may  not  be  so  simple  as  that  imposed 
by  compelling  the  moving  particle  to  remain  on  a  given  surface 
or  on  a  given  curve. 


10 


INTRODUCTION 


[Aitr.  6 


Take,  for  example,  the  tractrix  problem,  when  the  particle 
moves  on  a  smooth  horizontal  plane. 

Let  a  particle  of  mass  m,  attached  to  a  string  of  length  a,  rest 
on  a  smooth  horizontal  plane.  The  string  lies  straight  on  the 
plane  at  the  start,  and  then  the  end  not  attached  to  the  particle 
is  drawn  with  uniform  velocity  along  a  straight  line  perpendicu- 
lar to  the  initial  position  of  the  string  and  lying  in  the  plane. 
Let  us  take  as  our  coordinates  x,  the  distance  traveled  by 
that  end  of  the  string  which  is  not  attached  to  the  particle, 
and  0,  the  angle  made  by  the  string  with  its  initial  position. 
Let  R  be  the  tension  of  the  string  and  n  the  velocity  with 
which  the  end  of  the  string  is  drawn  along.  Let  X,  Y,  be  tho 
rectangular  coordinates  of  the  particle,  referred  to  the  i 
line  and  to  the  initial  position  of  the 
string  as  axes.  * 

X  =  x  —  a  sin  0, 
Y=  a  cos  0 1 
•     X=x  —  a  cos  66, 


r=- a  sin. 00. 


y 


cT 

-^7-  =  fn  \x  —  a  cos  00], 
ex 


—-  =  m  [a?0  —  a  cos  0x], 
c6 

— —  =  ma  sin  6x6, 
80 

',  —  \x  —  a  cos  00]  Bx  =  fi  sin  08z. 
ctt- 


m  —  (V0  —  a  cos  0x)  —  a  sin  6x6  \&d  =  0. 

K      W-tt, 
-il 


1 


CHAP.  I]     CONSTRAINED  MOTION  OF  A  PARTICLE          11 

Adding  the  condition  x  =  nt, 

and  reducing,      —  ma  [cos  00  —  sin  #02]  =  R  sin  0. 

ma20  =  0. 
0  =  0. 


Integrating,  0  =  C  =  -  • 


R  =  ma6\ 
h 

a 


The  particle  revolves  with  uniform  angular  velocity  about 
the  moving  center,  and  the  pull  on  the  string  is  constant. 

(b~)  A  particle  is  at  rest  in  a  smooth  horizontal  tube.  The 
tube  is  then  made  to  revolve  in  a  horizontal  plane  with  uniform 
angular  velocity  CD.  Find  the  motion  of  the  particle. 

Suggestion.  Take  the  polar  coordinates  r,  <£,  of  the  particle  as 
our  coordinates,  and  let  R  be  the  pressure  of  the  particle  on 
the  tube. 


Adding  the  condition  <£  =  wt^ 

and  reducing,  r  —  ofr  =  0, 

2  mwrr  =  Rr. 


m  [/•  -  r<j>2~\  Sr  =  0. 

d  ^° 

m 


12  INTRODUCTION  [ART.  6 

Solving,  r  =  A  cosh  cot  +  B  sinh  cot, 

r  =  a     and     r  =  0  at  the  start. 
Hence  r  =  a  cosh  <at  —  a  cosh  <£, 

R  =  2  ?naa>2  sinh  a>£  =  2  wao)2  sinh  <f>. 

If  we  are  interested  only  in  the  motions  and  not  in  the  reac- 
tions, problems  (a)  and  (J)  can  be  solved  more  simply.  If  in 
each  we  were  to  use  one  less  coordinate,  0  only  in  (a)  and  r 
only  in  (5),  rectangular  coordinates  X,  I7,  for  the  particle  could 
be  obtained  whenever  the  time  was  given,  and  therefore  could 
be  expressed  explicitly  in  terms  of  9  or  r  and  t.  A  careful 
examination  of  Art.  2  will  show  that  the  reasoning  is  extended 
easily  to  such  a  case,  and  that  the  work  done  by  the  effective 

forces  when  q1  only  is  changed  is  still    —  — —  -    S^V    It  i 

to  be  noted,  however,  that  when  the  rectangular  coordinates  are 
functions  of  t  as  well  as  of  q^  q2,  etc.,  the  energy  T  is  no  longer 
a  homogeneous  quadratic  in  q^  q2,  etc. 

(«')  X=nt  —  a  sin  0, 

Y=  acos#; 
X=  n  —  a  cos  6$, 


—  2  an  cos 


is 


BT 

—P  =  m  [a20  —  an  cos  01, 

de 

— -  =  man  sin  66. 
cv 

a^e  -  an  cos  (9) -  an  sin  66\  80  =  0. 


m 
\_cfr^ 

6  =  0. 


6  =  - ,  as  before. 
a 


.  I]     CONSTRAINED  MOTION  OF  A  PAETICLE          13 


dT 

-— 

or 


3r 

m  [r  — *o>V]  8r  =  0. 
r  =  a  cosh  o>£,  as  before. 

EXAMPLES 

1.  A  particle  rests  on  a  smooth  horizontal  whirling  table  and 
is  attached  by  a  string  of  length  a  to  a  point  fixed  in  the  table 
at  a  distance  b  from  the  center.  The  particle,  the  point,  and 
the  center  are  initially  in  the  same  straight  line.  The  table  is 
then  made  to  rotate  with  uniform  angular  velocity  &>.  Find  the 
motion  of  the  particle. 

Suggestion.  Take  as  the  single  coordinate  6  the  angle  made 
by  the  string  with  the  radius  of  the  point.  Let  X,  Y,  be  the 
rectangular  coordinates  of  the  particle,  referred  to  the  line  ini- 
tially joining  it  with  the  center  and  to  a  perpendicular  thereto 
through  the  center  as  axes. 

Then  X=b  cos  a>t  +  a  cos  (0  +  cof), 

v 
and  Y  =  b  sin  (ot  +  a  sin  (6  +  taf). 

T=^  [6V  +  a2  (w  +  0)2  +  2  aba)  cos  Q  (&>  +  0)], 


7      2 

and  6  -\  --  sin  0  =  0  ; 

a  "to 


,   ,  ,  ,     ,.  ,.  ,,        ,     ,,       .-V^o.      CX.         OAJi         KJ<YVv"«» 

and  the  relative  motion  on  the  table  is  simple  pendulum  motion, 
the  length  of  the  equivalent  pendulum  being  l  —  --^-- 

00) 

2.  A  particle  is  attracted  toward  a  fixed  point  in  a  horizontal 
whirling  table  with  a  force  proportional  to  the  distance.  It  is 
initially  at  rest  at  the  center.  The  table  is  then  made  to  rotate 


14  INTRODUCTION  [ART.  6 

with  uniform  angular  velocity  o>.    Find  the  path  traced  on  the 
table  by  the  particle. 

Suggestion.  Take  as  coordinates  x,  y,  rectangular  coordinates 
referred  to  the  moving  radius  of  the  fixed  point  as  axis  of 
abscissas  and  to  the  center  of  the  table  as  origin.  Let  X,  Y,  be 
the  rectangular  coordinates  referred  to  fixed  axes  coinciding 
with  the  initial  positions  of  the  moving  axes. 

X=  x  cos  cot  —  i/  sin  cot,          Y=x  sin  tat  +  y  cos  cat. 


Whence  come  m  [x  —  2  coy  —  coVj  =  —  /i  (x  —  a), 
m  [y  +  2  cox  —  a?y~\  =  —  py. 

If  a)2  =  .c  ,  the  solution  is  easy  and  interesting. 
m 

I  <x  j  x—  2  o>y  =  a«o2,  (1) 

y  +  2  mx  =  0.  (2) 

Integrating  (2),  y  +  2  cox  =  0. 

Substituting  in  (1),    X  +  4  afx  =  aa>2.  (3) 

Multiplying  (3)  by  2i,  and  integrating, 
s?  +  4  w  V  =  2  a«2a:. 


Whence 


#  =  —  [1  —  cos  2  art], 
y  =  —  -  [2  cot  —  sin  2 
Replacing  2  cot  by  0,  x=-\\—  cos  ^], 


CHAP.  I]     MOTION  OF  A  SYSTEM  OF  PARTICLES  15 

and  the  curve  traced  on  the  table  is  the  cycloid  generated  by  a 
circle  of  radius  —  rolling  backward  along  the  moving  axis  of  Y. 

7.  A  System  of  Particles.  If  instead  of  a  single  particle  we 
have  a  system  of  particles,  free,  or  connected  or  otherwise  con- 
strained, m'x,  my,  mi,  are  the  effective  forces  on  the  particle  P. 
The  effective  forces  on  all  the  particles  are  spoken  of  as  the 
effective  forces  on  the  system  and  are  mechanically  equivalent 
to  the  set  of  actual  forces  on  the  system. 

T,  the  kinetic  energy  of  the  system,  is  the  sum  of  the  kinetic 
energies  of  all  the  particles. 


If  8 W  is  the  work  done  by  the  effective  forces  in  any  supposed 
infinitesimal  displacement  of  the  system, 

&W=  Sm  [xSx  +  yfy  +  zSz]. 

If  the  particles  of  a  moving  system  are  subjected  to  connec- 
tions or  constraints,  these  connections  or  constraints  may  or 
may  not  vary  with  the  time.  In  the  latter  caje  a  set  of  any  n 
independent  variables  q^  q2,  •  •  •,  qn,  such  that  when  they  and 
the  connections  and  constraints  are  given,  the  position  of  every 
particle  of  the  system  is  uniquely  determined,  and  such  that 
when  the  positions  of  all  the  particles  of  the  system  are  given, 
<?i'  <?»'  '  '  '•>  ?»»  follow  uniquely,  may  be  taken  as  coordinates  of 
the  system ;  and  n  is  called  the  number .  of  degrees  of  freedom 
of  the  system. 

In  the  former  case  a  set  of  variables  q^  q2,  •  •  •,  qn,  such  that 
when  they  and  the  time  are  given,  the  position  of  every  particle 
of  the  system  is  uniquely  determined,  and  such  that  when  the 
positions  of  all  the  particles  of  the  system  and  the  time  are 
given,  q^  q^  • .  .,  qn,  follow  uniquely,  may  be  taken  as  the  coor- 
dinates of  the  system ;  and  n  is  called  the  number  of  degrees  of 
freedom  of  the  system. 


16  INTRODUCTION  [ART.  7 

The  equations  expressing  the  connections  and  constraints  in 
terms  of  the  rectangular  coordinates  of  the  particles  and  of  the 
coordinates  q^  qz,  •  •  •,  qn,  of  the  system  are  often  called  the 
geometrical  equations  of  the  system  and  may  or  may  not  contain 
the  time  explicitly.  In  the  latter  case  the  geometrical  equations 
make  it  possible  to  express  the  coordinates  x,  y,  z,  of  every 
point  of  the  system  explicitly  as  functions  of  the  q*s  ;  in  the 
former  case,  as  functions  of  t  and  the  q's. 

The  geometrical  equations  must  not  contain  explicitly  either 
the  time  derivatives  of  the  rectangular  coordinates  of  the  parti- 
cles or  those  of  the  coordinates  q^  qz,  •  •  •,  <?B,  of  the  system 
unless  they  can  be  freed  from  these  derivatives  by  integration. 

Examples  of  geometrical  equations  not  containing  the  time 
explicitly  are  the  formulas  for  transformation  of  coordinates  hi 
Arts.  1  and  3,  and  the  equations  for  X  and  Y  in  Art.  6,  (a). 

Geometrical  equations  containing  the  time  are  the  equations 
for  X  and  Y  in  Art.  6,  (a'),  and  in  Art.  6,  Exs.  1  and  2. 

The  work,  8q  W,  done  by  the  effective  forces  when  ql  is 
changed  by  8q^  without  changing  the  other  q's  is  proved  to  be 


by  reasoning  similar  to  that  used  in  Art.  2.  For  the  sake  of 
variety  we  take  the  case  where  the  geometrical  equations 
involve  the  tune. 

Here  x=f[t,  qf  qz,  •  •  -,  ?„]. 

dx      dx  .        Bx  .  ex  . 

v  =  -^  +  —  q1  +  ^-q..-\  -----  1-  T—  qn, 

ct      cq^      dq^1-  dqm* 

and  is  an  explicit  function  of  t,  q^  q^  •  •  -,  qn,  q^  qz,  •  •  •,  qn. 

dx       dx 


d  dx       dzx       C2x  .         czx  c~. 

and  since    —  —  =  —  —  +  —  q  +  —  —  q  H  -----  h  :—  7—  qn, 
dtcch 


CHAP.  J]     MOTION  OF  A  SYSTEM  OF  PARTICLES  17 

dx       tfx       d*x  .         c*x  <?x 


and 


d  (dx\      ex 

-j.   5—  1  =  7—  • 
dt\dqj      o9l 


_ 


x 

and  therefore  fi  ^F  =    —  (  —  -    -  —  \6q  , 


and  if  Q18q1  is  the  work  done  by  the  actual  forces  when  q1  is 
changed  by  Sqf  d  dT      dT 

~T~    ~  --  ~  -  ==     ^1  '  \       J 

at  oqv      cq1 

If  the  geometrical  equations  do  not  contain  the  time,  the  same 
result  is  seen  to  hold,  and  in  this  case  it  is  to  be  noted  that 
since  x  is  homogeneous  of  the  first  degree  in  the  time  derivatives 
of  the  coordinates,  that  is,  in  q^  <?2,  •  •  -,  qn,  the  kinetic  energy  T 
is  a  homogeneous  quadratic  in  q^  qz,  •  •  -,  qn. 

Generally  every  one  of  the  n  equations  of  which  equation  (3) 
is  the  type  will  contain  all  the  n  coordinates  q^  q.2,  •  •  •,  qn,  and 
their  time  derivatives,  and  can  be  solved  only  by  aid  of  the 
others.  That  is,  we  shall  have  n  coordinates  and  the  time  con- 
nected by  n  simultaneous  differential  equations  no  one  of  which 
can  ordinarily  be  solved  by  itself. 

If  the  forces  exerted  by  the  connections  and  constraints  do 
no  work,  they  will  not  appear  in  our  differential  equations.^ 
Should  we  care  to  investigate  any  of  them,  we  have  only  to 
suppose  the  constraints  in  question  removed  and  the  number 
of  degrees  of  freedom  correspondingly  increased,  and  then  to 
replace  the  constraints  by  the  forces  they  exert  and  to  form 
the  full  set  of  equations  on  the  new  hypothesis. 


18  INTRODUCTION  [Am.  8 

8.  A  System  of  Particles.  Illustrative  Examples,  (a)  A  rough 
plank  16  feet  long  rests  pointing  downward  on  a  smooth  plane 
inclined  at  an  angle  of  30°  to  the  horizon.  A  dog  weighing 
as  much  as  the  plank  runs  down  the  plank  just  fast  enough 
to  keep  it  from  slipping.  What  is  his  velocity  when  he  reaches 
its  lower  end  ? 

Here  we  have  two  degrees  of  freedom.  Take  x,  the  distance 
of  the  upper  end  of  the  plank  from  a  fixed  horizontal  line  in 
the  plane,  and  ?/,  the  distance  of  the  dog  from  the  upper  end 
of  the  plank,  as  coordinates,  and  let  R  be  the  backward  force 

exerted  by  the  dog  on  the  plank,  and  m  the  \B9Jgfrt  of  the  dog. 

• 


dT 

—  =  m 

dT 

—  =  m 

,  m  [2  x  +  y"}  8x  =  2  mg  sin  30°  Sz. 
m  [x  +  y"]  by  =  IB  +  mg  sin  30° 


By  hypothesis,  x  =  a  constant, 

and  therefore  y  =  g, 


When  y  —  16,  y  =  32,  nearly. 

R      q 

Since  v  =  —  \-^i 

m      2 


CHAP.  I]     MOTION  OF  A  SYSTEM  OF  PARTICLES 


19 


(5)  A  weight  4m  is  attached  to  a  string  which  passes  over 
a  smooth  fixed  pulley.  The  other  end  of  the  string  is  fastened 
to  a  smooth  pulley  of  weight  m,  over  which  passes  a  second 
string  attached  to  weights  m  and  2m. 
The  system  starts  from  rest.  Find  the 
motion  of  the  weight  4m. 

Two  coordinates,  x,  the  distance  of  4m 
below  the  center  of  the  fixed  pulley,  and 
y,  the  distance  of  2m  below  the  center 
of  the  movable  pulley,  will  suffice.  The 
velocities  are  .  „  t  . 


4m 


x  for  4  m, 
—  x  for  movable  pulley, 

—  x  +  y  for  2  m, 

—  x  —  y  for  m. 

T  =  i  [4  mx2  +  mx2  +  2  m  (ij  —  x)2  +  m  (x  +  y 


|      |     2m 


dT 

_  =  w[8z-^], 

dT 

—  =  w[3^-fl. 

m  [8  x  —  y"\  Bx  =  [4  mg  —  mg  —  2  mg  —  mg~\  8x. 
m  [3  y  —  x~\  8y  =  [2  wy  — 

8  x  -  y  =  0. 


-- 
23 


The  weight  4  m  will  descend  with  uniform  acceleration  equal 
to  one  twenty-third  the  acceleration  of  gravity. 

(c)  The  dumb-bell  problem.  Two  equal  particles,  each  of  mass 
m,  connected  by  a  weightless  rigid  bar  of  length  a,  are  set 


20  INTRODUCTION  [ART.  8 

moving  in  any  way  on  a  smooth  horizontal  plane.  Find  the 
subsequent  motion. 

We  have  three  degrees  of  freedom.  Let  x,  y,  be  the  rectangu- 
lar coordinates  of  the  middle  of  the  bar,  and  6  the  angle  made 
by  the  bar  with  the  axis  of  X. 

The  rectangular  coordinates  of  the  two  particles  are 

a  a    •     a  T 

x  —  ^costf,  2/--sm0       and 

Z  A 

their  velocities  are 


and  -*!  ( x  —  -  sin  00 j  +  ( y  +  -  cos  i 

m\  a2  • 

T=  —  \j?  +  f  +  —  0'2  +  (asm0-acos0)0(x  +  y') 

+  x*  +  y2  +  7-  #2  4-  («  cos  ^  —  a  sin  6*)6(x- 


2  mz&c  =  0, 
2  my^y  =  0, 

o 
Wlft        •• 

~  080  =  0. 


=  0. 


CHAP.  I]    MOTION  OF  A  SYSTEM  OF  PAETICLES  21 

Hence  the  middle  of  the  bar  describes  a  straight  line  with 
uniform  velocity,  and  the  bar  rotates  with  uniform  angular 
velocity  about  its  moving  middle  point. 

EXAMPLE 

Two  Alpine  climbers  are  roped  together.  One  slips  over  a 
precipice,  dragging  the  other  after  him.  Find  their  motion 
while  falling. 

Ans.  Their  center  of  gravity  describes  a  parabola.  The  rope 
rotates  with  uniform  angular  velocity  about  their  moving  center 
of  gravity. 

(cf)  Two  equal  particles  are  connected  by  a  string  which 
passes  through  a  hole  in  a  smooth  horizontal  table.  The  first 
particle  is  set  moving  on  the  table,  at  right  angles  with  the 
string,  with  velocity  v  'ag  where  a  is  the  distance  of  the  particle 
from  the  hole.  The  hanging  particle  is  drawn  a  short  distance 
downward  and  then  released.  Find  approximately  the  subse- 
quent motion  of  the  suspended  particle. 

Let  x  be  the  distance  of  the  second  particle  below  its  position 
of  equilibrium  at  the  time  t,  and  6  the  angle  described  about 
the  hole  in  the  time  t  by  the  first  particle. 


dT 

— 

dx 

8T 

C/  JL  ,  ^     An 

—  =  —  mfa  —  X\Q\ 

ex 
fiT 

v  -1-  s  i^  9/1 

—  =  m  (a  —  x)"V. 
c6 

01  [2  *  +  (a  -  a:)  02]  Sx  =  mgSx,  (1) 

™|[(a-*)20]S0  =  0.  (2) 

2£  +  (a-z)02  =  g.  (3) 

(a  -  x^B  =  C=a  Vag,  (4) 


22  .'  INTRODUCTION  [ART.  9 

since  (2)  holds  good  while  the  hanging  particle  is  being  drawn 
down  as  well  as  after  it  has  been  released. 


# 
2  x  H  --  #  =  0,  approximately, 


and  £  +  -    a;  =  0. 

2a 

For  small  oscillations  of  a  simple  pendulum  of  length 


Therefore  the  suspended  particle  will  execute  small  oscilla- 
tions, the  length  of  the  equivalent  simple  pendulum  being  |  a. 

EXAMPLE 

A  golf  ball  weighing  one  ounce  and  attached  to  a  strong 
string  is  "  teed  up  "  on  a  large,  smooth,  horizontal  table.  The 
string  is  passed  through  a  hole  in  the  table,  10  feet  from  the 
ball,  and  fastened  to  a  hundred-pound  weight  which  rests  on  a 
prop  just  below  the  hole.  The  ball  is  then  driven  horizontally, 
at  right  angles  with  the  string,  with  an  initial  velocity  of  a 
hundred  feet  a  second,  and  the  prop  on  which  the  weight  rests 
is  knocked  away. 

(a)  How  high  must  the  table  be  to  prevent  the  weight  from 
falling  to  the  ground?  (&)  What  is  the  greatest  velocity  the 
golf  ball  will  acquire  ?  Ans.  (a)  8.96  ft.  (b)  963.4  ft.  per  sec. 

9.  Rigid  Bodies.  Two-dimensional  Motion.  If  the  particles  of 
a  system  are  so  connected  that  they  form  a  rigid  body  or  a 
system  of  rigid  bodies,  the  reasoning  and  formulas  of  Art.  7 
still  hold  good. 


CHAP.  I]         PLANE  MOTION  OF  KIGID  BODIES  23 

(«)  Let  any  rigid  body  containing  a  horizontal  axis  fixed  in 
the  body  and  fixed  in  space  move  under  gravity.  Suppose  that 
the  body  cannot  slide  along  the  axis.  Then  the  motion  is 
obviously  rotational,  and  there  is  but  one  degree  of  freedom. 
Take  as  the  single  coordinate  the  angle  0  made  by  a  plane  con- 
taining the  axis  and  the  center  of  gravity  of  the  body  with  a 
vertical  plane  through  the  axis. 

Let  h  be  the  distance  of  the  center  of  gravity  from  the  axis, 
and  k  the  radius  of  gyration  of  the  body  about  a  horizontal 
axis  through  the  center  of  gravity.  Then 


T  =     (A2  +  &2)  e\    (v.  App.  A,  §  §  5  and  10) 


m  (A2  +  F)  880  =  -  mgh  sin  080. 


and  we  have  simple  pendulum  motion,  the  length  of  the  equiv- 
alent simple  pendulum  being 

i— * 


h 

(5)  Two  equal  straight  rods  are  connected  by  two  equal 
strings  of  length  a  fastened  to  the  ends  of  the  rods,  the  whole 
forming  a  quadrilateral  which  is  then  suspended  from  a  hori- 
zontal axis  through  the  middle  of  the  upper  rod.  The  system 
is  set  moving  in  a  vertical  plane.  Find  the  motion. 

Take  as  coordinates  <£,  the  inclination  of  the  upper  rod  to 
the  horizon,  and  0,  the  angle  made  with  the  vertical  by  a  line 
joining  the  point  of  suspension  with  the  middle  of  the  lower 
rod.  From  the  nature  of  the  connection  the  rods  are  always 
parallel. 

Let  k  be  the  radius  of  gyration  of  each  rod  about  its  center 
of  gravity. 


24  INTRODUCTION  [ART.  9 

(v.  App.  A,  §  10) 


—  =  ma  0. 
80. 


=  0, 


and  the  rods  revolve  with  uniform  angular  velocity  while  the 
middle  point  of  the  lower  rod  is  oscillating  as  if  it  were  the 
bob  of  a  simple  pendulum  of  length  a. 

(c)  If  an  inclined  plane  is  just  rough  enough  to  insure  the 
rolling  of  a  homogeneous  cylinder,  show  that  a  thin  hollow 
drum  will  roll  and  slip,  the  rate  of  slipping  at  any  instant  being 
one  half  the  linear  velocity. 

Let  x  be  the  distance  the  axis  of  the  cylinder  has  moved 
down  the  incline,  6  the  angle  through  which  the  cylinder  has 
rotated,  a  the  radius  of  the  cylinder,  and  a  the  inclination  of 
the  plane.  Call  the  force  of  friction  F. 


8T 

—  =  rn-x. 

dx 


7. 
—  r  =  mnt. 

cB 
mxSx  =  [mg  sin  a  —  F~\  Sx, 


If  there  is  no  slipping,  x  =  a0, 

mx  =  mg  sin  a  —  F, 


CHAP.  I]        PLANE  MOTION  OF  KIGLD  BODIES  25 


Fa* 
Hence  —^-  =  mg  sin  a—F, 


a? 
For  a  solid  cylinder,     k2  =  — , 

25 

^  =  -g-  w  </  sin  a ; 
Ji  =  mg  cos  a, 

where  K  is  the  pressure  on  the  plane-, 

F      1 

,-5-ji^ 

wiiere  /A  is  the  coefficient  of  friction. 
For  a  hollow  drum,       k  =  a, 

F  =  i  »?$r  sin  a, 

77T  ~| 

^  =  2 

/u,  <  i  tan  a, 

and  the  drum  will  slip. 
For  the  drum,  then, 

F  =  pR  =  fj,mg  cos  a  =  ±mg  sin  a, 


mx  =  mg  sin  a  —  -1  7/^7  sin  «  =  f  ^  sin  or, 
i  =  I-  <T£  sin  or. 


#2$  =  Fa  =  -  mg  sin  a, 
o 

aO  =  ^g  sin  a, 
a0  =  lt  sin  a. 


S  =  x  —  a6  =  —  mgt  sin  a  =  —  •, 
o  2 


where  S  is  the  rate  of  slipping. 


26  INTRODUCTION  [ART.  10 

EXAMPLES 

f 

1.  A  sphere  rotating  about  a  horizontal  axis  is  placed  on  a 
perfectly  rough  horizontal  plane  and  rolls  along  in  a  straight 
line.    Show  that  after  the  start  friction  exerts  no  force. 

2.  A  sphere  starting  from  rest  moves  down  a  rough  inclined 
plane.     Find   the   motion,     (a)  What  must  the  coefficient   of 
friction  be  to  prevent  slipping  ?     (5)  If  there  is  slipping,  what 
is  its  velocity  ? 

Ans.  (a)  yu>|tana:.      (5)   $=</£[sin  a  —  £/i  cos  a]. 

3.  A  wedge  of  mass  M  having  a  smooth  face  and  a  perfectly 
rough  face,'  making  with  each  other  an  angle  a,  is  placed  with  its 
smooth  face  on  a  horizontal  table,  and  a  sphere  of  mass  m  and 
radius  a  is  placed  on  the  wedge  and  rolls  down.   Find  the  motion. 

Let  x  be  the  distance  the  wedge  moves  on  the  table,  and  y 
the  distance  the  sphere  rolls  down  the  plane. 

Note  that    T  =  i  [M+  m]x2  +  ^  K  *  **  if-^xy  cos  a\ . 
Ans.   (m  +  M )  x  —  my  cos  a  =  0,     |-  y  —  x  cos  a  =  ^  gt2  sin  a. 

10.  Rigid  Bodies.  Three-dimensional  Motion.  («)  A  homo- 
geneous sphere  is  set  rolling  in  any  way  on  a  perfectly  rough 
horizontal  plane.  Find  the  subsequent  motion. 

Let  x,  y,  #,  be  the  coordinates  of  the  center  of  the  sphere 
referred  to  a  set  of  rectangular  axes  fixed  in  space;  two  of 
which,  the  axes  of  X  and  r,  lie  in  the  given  plane.  Let  OA, 
OB,  OC,  be  rectangular  axes  fixed  in  the  sphere  and  passing 
through  its  center ;  let  OX,  0  Y,  OZ,  be  rectangular  axes  through 
the  center  of  the  sphere  parallel  to  the  axes  fixed  in  space;  and 
let  0,  <£,  ty  be  the  Euler's  angles  (v.  App.  A,  §  8).  Take  x,  y, 
6,  <£>,  and  ty  as  our  coordinates.  The  only  force  we  have  to 
consider  is  F,  the  friction,  and  we  shall  let  Fx  and  Fy  be  its 
components  parallel  to  the  axes  OX,  OY,  respectively. 


CHAP.  I]      MOTION  OF  RIGID  BODIES  IN  SPACE               27 
where                   (ox  =  —  0  sin  i/r  4-  <£  sin  0  cos  i^-, 

Wy  =  0  cos  A/T  4-  </>  sin  0  sin  i^-, 

<•>,  =  <$>  cos  0  +  ^.  (v.  App.  A,  §  8) 

Hence  T  =  —  [i2  4-  y2  4-  A:2 (02  4-  <£2  4- 1^2  4-  2  cos  0^-^)]. 

We  get                               wi  =  Fx,  (1) 

my  =  Fv,  (2) 

wF  -^  [-f  +  cos  0^]  =  0,  (3) 


—  [^  +  cos  0-^]  =  —  a^x  sin  0  sin  i/r  +  aJ^y  sin  0  cos  ^,     (4) 
at 

mk2  [0  +  sin  0</>-^]  =  —  a^x  cos  i/r  —  a^  sin  i|r  ;  (5) 

and  as  there  is  no  slipping, 

x  —  acoy  =  x  —  a  (0  cos  i/r  +  ^  sin  0  sin  i|r)  =  0,  (6) 

y  +  a&>x  =  y  +  a  (—  0  sin  i/r  +  ^  sin  0  cos  i/r)  =  0.        (7) 
From  (4)  and  (5), 


k?  [sin  T/T  —  (<£  +  cos  0-^)  +  sin  0  cos  i/r  (0  -f  sin 


mk2  [cos  T/T  —  (^  +  cos  0-^)  —  sin  0  sin  i/r  (^  -f  sin 


(8) 


=  aJ?;  sin  0.        (9) 

Expanding  the  first  members  of  (8)  and  (9)  and  eliminat- 
ing -^r  by  the  aid  of  (3),  we  get 


ml?  [cos  i/T0  —  sin  ->|r0^r  +  sin  0  sin  -»/r<£  4-  cos  0  sin 

4-  sin  0  cos  ^0-^]  =  -  a^,         (10) 


mk2  [—  sin  -\/r0  —  cos  ^^  4-  sin  0  cos  i/r<£  4-  cos  0  cos 

—  sin  0  sin  i/r<^r]  =  a^.  (11) 


28  INTRODUCTION  [ART.  10 

- 
at 


But  the  first  members  of  (10)  and  (11)  are  obviously  mk2  - 


•j 
and  mtf  -=-*,  respectively.    Hence,  by  (6)  and  (7), 

(AJV 

mtf  .. 

-  x=-aFx, 


mk*  .. 

•—y=aF>- 

Substituting  in  (1)  and  (2),  we  get 

mk2  .. 

-  x  =  —  max, 
a 


whence 


From  (7)  and  (6),  =  0, 

dt 


dt 
and  from  (3),  ^  =  0. 

Cvv  ' 

Finally,  Fx  =  0, 

7^  =  0. 

Hence  the  center  of  the  sphere  moves  in  a  straight  line  with 
uniform  velocity,  and  the  sphere  rotates  with  uniform  angular 
velocity  about  an  instantaneous  axis  which  does  not  change  its 
direction,  and  no  friction  is  brought  into  play  after  the  rolling 
begins. 

(6)  The  billiard  ball.  Suppose  the  horizontal  table  in  (a)  is 
imperfectly  rough,  coefficient  of  friction  ^,  and  suppose  the 
ball  to  slip. 

Take  the  same  coordinates  as  before,  and  equations  (1),  (2), 
(3),  (4),  (5),  (8),  (9),  (10),  and  (11)  still  hold  good.  Let  a 


CHAP.  I]  THE  BILLIARD  BALL  29 

be  the  angle  the  direction  of  the  resultant  friction,  F  —  pmcf, 
makes  with  the  axis  of  X,  and  let  S  be  the  velocity  of  slip- 
ping, that  is,  the  velocity  with  which  the  lowest  point  of  the 
ball  moves  along  the  table.  Of  course  the  directions  of  F  and  S 
are  opposite. 

Let  Sx  and  Sy  be  the  components  of  S  parallel  to  the  axes  of 
X  and  F.    We  have 

—  S  cos  a  =  Sx  =  x  —  acoy , 
and  —  S  sin  a  =  Sy  =  y  +  aa>x. 

Fx  =•  /j,mg  cos  or, 
and  Fy  =  \trng  sin  a. 

dSr  da  dS  da>,, 

=  S  sin  a  — —  cos  a—-  =  x  — 


dt  dt  dt  dt 

dSu  da       .       dS      ..  ,      oV. 

— K  =  —  S  cos  a- — —  sm  a  —  =  w  +  a  -— £ . 
dt  dt  dt  dt 

From  (1),  x  =  /*#  cos  a, 

and  from  (10),  — ^  =  —  —  pg  cos  a. 

Cv(/  K 

Hence         S  sin  a cos  a  —  =  — — —  fig  cos  or, 

dt  dt          k 

and  from  (2)  and  (11), 

da       .       dS     az+~k? 

—  S  cos  a  — —  sin  a  -—  =  — — —  fig  sin  a.  (13) 

dt  dt         KT 

Multiplying  (12)  by  sin  a  and  (13)  by  cos  a,  and  subtracting, 

S—  =  0.  (14) 

dt 

Multiplying  (12)  by  cos  a  and  (13)  by  sin  a,  and  adding, 
dS     a2  +  k2  .,  .. 

-^=^^w-  (15) 


30  INTEODUCTION  [ART.  10 


Integrating  (15),     S  =  S0  -    ~  iqt.  (16) 

From  (14),  a  =  «o, 

and  the  direction  of  slipping  does  not  change. 

If  the  axes  are  so  chosen  that  the  axis  of  X  has  the  direction 
opposite  to  the  direction  of  slipping,  a  =  0.    Then 


These  equations  are  familiar  in  the  theory  of  projectiles, 
and  the  path  traced  on  the  table  is  a  parabola  so  long  as 
slipping  lasts. 

Should  i/Q  happen  to  be  zero,  the  path  degenerates  into  a 
straight  line. 

When  slipping  stops, 

x  —  aa>y  =  0,     and     y  +  aa)x  =  0, 
and  we  have  the  case  treated  in  (a). 

EXAMPLE 

A  homogeneous  sphere  is  set  rolling  on  a  perfectly  rough 
inclined  plane.  Find  the  path  traced  on  the  plane. 

Ans.   A  parabola. 

(<?)  The  gyroscope.  Suppose  a  rigid  body  containing  a  fixed 
point  and  having  two  of  its  moments  of  inertia  about  its  prin- 
cipal axes  through  the  fixed  point  equal.  Obtain  the  differ- 
ential equations  for  its  motion  under  gravity. 

We  shall  use  Euler's  angles  with  a  vertical  axis  of  Z. 

We  have  col  =  6  sin  <£  —  -^r  sin  0  cos  (/>, 

w^—Q  cos  </>  +  ^  sin  6  sin  <£>, 
o)3  =  ^r  cos  6  +  <j>.  (v.  App.  A,  §  8) 

T  =  \  [Aa>?  +  Au>l  +  <7«32]  (v.  App.  A,  §  10) 

sin2  O    2  +  Cir  cos 


CHAP.  I]  THE  GYROSCOPE  31 

8T 

1±-  =  C  0/r  COS  6  +  </>), 


—r  =A  sin2  6     +  ^cos  0(     cos  (9  +  </>), 


cd 

f1  T 


Our  equations  are    C—  (i/r  cos  6  +  <£)  =  0,  (1) 

wv 


=A  sin  0  cos  <ty2  -  (7  sin  0  (^  cos  0  + 


0,  (2) 

Ct 

Ad  —  ^4  sin  ^  cos  6tyz  +  C  sin  ^  (-^  cos  0  +  </>)  -^  =  ?n#a  sin  ^.  (3) 

From  (1),  ^  cos  ^  +  (j>  —  a,  (4) 

where  a  is  the  initial  velocity  about  the  axis  of  unequal  moment. 

A  sin2  6ty  +  Ca  cos  0  =  £.  (5) 

J61'  —  A  sin  0  cos  6^  +  (7#  sin  0-^  =  mga  sin  0,  (6) 

or  substituting  ^jr  from  (5), 

•:a      (L  —  Cacos0~)(Lcos0  —  Co) 
v  -  >v    -  y 


sm 
sm 

(c?)  Obtain  Euler's  equations  for  a  rigid  body  containing  a 
fixed  point. 

Here  T  =  \  [^w2  +  Ba>%  +  <X].         (v.  App.  A,  §  10) 

dT 

-7-==^..  (v.  App.  A,  §8) 

#<p 

O  /T7 

•—  -  =Ao)1  [0  cos  </>  +  ^  sin  0  sin  <£] 

[—  0  sin  $  +  ^  sin  6  cos  0] 


32  INTKODUCTION  [ART.  11 

Whence 


where  N  is  the  moment  of  the  impressed  forces  about  the  C  axis. 


. 

The  remaining  two  Euler's  equations  follow  at  once  from 
this  by  considerations  of  symmetry. 

11.  In  Arts.  2  and  7  it  was  shown  that  under  slight  limi- 
tations the  coordinates  of  a  moving  particle  or  of  a  moving  sys- 
tem could  be  taken  practically  at  pleasure,  and  the  differential 
equations  of  motion  could  be  obtained  by  the  application  of  a 
single  formula.  It  does  not  follow,  however,  that  when  it  comes 
to  solving  a  concrete  problem  completely,  the  choice  of  coordi- 
nates is  a  matter  of  indifference.  Different  possible  choices 
may  lead  to  differential  equations  differing  greatly  in  compli- 
cation, and  as  a  matter  of  fact  in  the  illustrative  problems  of 
the  present  chapter  the  coordinates  have  been  selected  with  care 
and  judgment.  That  this  care,  while  convenient,  is  not  essential 
may  be  worth  illustrating  by  a  practical  example,  and  we  shall 
consider  the  simple  familiar  case  of  a  projectile  in  vacua. 

Altogether  the  simplest  coordinates  are  x  and  y,  rectangular 
coordinates  referred  to  a  horizontal  axis  of  X  and  a  vertical 
axis  of  Y  through  the  point  of  projection. 


We  have 


dT 

—• 

ox 

dT 


mx=  0, 

my  =  —  mg. 


CHAP.  I]  THE  CHOICE  OF  COORDINATES  33 

Solving,  x  =  vx, 


x  =  vxt, 

y=V+y- 

Let  us  now  try  a  perfectly  crazy  set  of  coordinates,  q1  and 

where 

ql~x-\-  tan  ly, 

and  q2  =  x  —  tan~  l  y. 

Proceeding  in  our  regular  way,  we  have 


4    ,  —    »  , 

—  =  -  sec4  ^-^  tan  * 


m       4  &  —  o.2  ,       q^—  Q*  , 
—  =  -  -  sec*  .fcA  tan  A_i 


LZ     tan 
—  f 


-  sec 


INTRODUCTION 


"  "  \  4  <1  ~     2  i 

~       "  S  ^ 


[ART.  11 

• 

?1  "  ?a)  J 
.        (2) 


Adding  (1)  and  (2),   —  (^  -+-  <j2)  =  0. 
Whence  ^  +  qz  =  2 1^, 

and  <?i  +  qa  =  2  vxt 

Subtracting  (2)  from  (1), 


(3) 


=  —  mg  sec2  ^J 


4 


Multiplying  by  —  (^  —  ^2),  and  integrating, 

7/fr 

sec*  2Lii       -        =  -  8  tan        ^ 


Let 


sec2  z  —  =  v  v^  —  2  ^7  tan  2, 


- 
V  t>J  —  2  <?  tan 


tan 


B--Je  =  tan-1lt;J-^|. 

» 


CHAP.  I]  NOMENCLATURE  35 

But  from  (3),     ^J±^  =  vj, 

A 

Hence  q1  =  vxt  +  tan"  1  \vyt  —  ^—  , 


Of  course  this  should  agree  with  our  first  answer,  and  a 
moment's  consideration  shows  that  it  does. 

We  have  gl  \  q'2  =  x  =  vxt, 

A 

q,  —  q.-,  at2          ,    f 

tan  31      •"  =  y  =  vyt  —  -—  ,  as  before. 

A  A 

12.  The  parameters  q^  qz,  •  •  •  that  we  have  been  using  to 
fix  the  position  of  our  moving  particle  or  moving  system  are 
called  generalized  coordinates.  Following  the  analogy  of  rectan- 
gular coordinates,  the  time  derivative  qk  of  any  generalized 
coordinate  qk  is  called  the  generalized  component  of  velocity 
corresponding  to  qk.  It  may  be  a  linear  velocity,  or  an  angular 
velocity  as  in  many  of  our  problems,  or  it  may  be  much  more 
complicated  than  either  as  in  our  latest  example. 

The  kinetic  energy  T  expressed  in  terms  of  the  generalized 
coordinates  and  the  generalized  velocities  is  called  the  Lagran- 
gian  expression  for  the  kinetic  energy. 

If  we  are  using  rectangular  coordinates  and  dealing  with  a 
moving  particle, 


dx 

and  is  the  x  component  of  the  momentum  of  the  particle. 

cT 
Following  this  analogy,  ^-7-   is   called  the  generalized  com- 

°<lk 

ponent  of  the  momentum  of  the  system,  corresponding  to  the 
coordinate  qt.   It  is  frequently  represented  by  pt,  and  may  be  a 


36  INTRODUCTION  [ART.  13 

momentum,  or  a  moment  of  momentum  as  in  many  of  our  prob- 
lems, or  it  may  be  much  more  complicated  than  either  as  in 
our  latest  example. 
Equations  of  the  type 

dt  dqk      dqk 

are  practically  what  are  called  the  Lagrangian  equations  of 
motion,  although  strictly  speaking  the  regulation  form  of  the 
Lagrangian  equations  is  a  little  more  compact  and  will  be 
given  later,  in  Chapter  IV. 

Qt,  defined  through  the  property  that  Qk§qk  is  the  work  done 
by  the  actual  forces  when  qk  is  changed  by  Sqk,  is  called  the 
generalized  component  of  force  corresponding  to  qk.  It  may  be 
a  force,  or  the  moment  of  a  force  as  in  many  of  our  problems, 
or  it  may  be  much  more  complicated  than  either  as  in  our 
latest  example. 

13.  Summary  of  Chapter  I.  If  a  moving  system  has  a  finite 
number  n  of  degrees  of  freedom  (v.  Art.  7)  and  n  independent 
generalized  coordinates  q^  q^  •  •  •,  qn,  are  chosen,  the  kinetic 
energy  T  can  be  expressed  in  terms  of  the  coordinates  and 
the  generalized  velocities  q^  j2,  •  •  •,  qn,  and  when  so  expressed 
will  be  a  quadratic  in  the  velocities,  a  homogeneous  quad- 
ratic if  the  geometrical  equations  (v.  Art.  7)  do  not  contain  the 
time  explicitly. 

The  work  done  by  the  effective  forces  in  a  hypothetical  infin- 
itesimal displacement  of  the  system  due  to  an  infinitesimal 
change  dqk  in  a  single  coordinate  qk  is 


dtdqk      dqk 

If  this  is  written  equal  to  Qk$qk,  the  work  done  by  the  actual 
forces  in  the  displacement  in  question,  there  will  result  the 
Lagrangian  equation  ,  -  - 

dt  c(jk      cqk 


CHAP.  I]  SUMMARY  87 

The  n  equations  of  which  this  is  the  type  form  a  set  of  simul- 
taneous differential  equations  of  the  second  order,  connecting 
the  n  generalized  coordinates  with  the  time.  When  the  complete 
solution  of  this  set  of  equations  has  been  obtained,  the  problem 
of  the  motion  of  the  system  is  solved  completely. 

It  must  be  kept  in  mind  that  in  order  to  obtain  the  value  of 
a  single  coordinate  or  of  a  set  of  coordinates  less  in  number 
than  n  it  is  generally  necessary  to  form  and  to  solve  the  com- 
plete set  of  n  differential  equations. 

We  shall  see,  however,  in  the  next  chapter,  that  in  certain 
important  classes  of  problems  some  of  these  equations  need  not 
be  formed,  and  that  some  of  the  coordinates  can  be  safely 
ignored  without  interfering  with  our  obtaining  the  values  of  the 
remaining  coordinates ;  that,  indeed,  we  may  be  able  to  handle 
satisfactorily  some  problems  concerning  moving  systems  having 
an  infinite  number  of  degrees  of  freedom. 

44922 


CHAPTER  II 

THE  HAMILTONIAN  EQUATIONS.    ROUTH'S  MODIFIED 

LAGRANGIAN  EXPRESSION.    IGNORATION  OF 

COORDINATES 

14.  The  Hamiltonian  Equations.  If  the  geometrical  equations 
of  the  system  (v.  Art.  7)  do  not  contain  the  time  explicitly 
and  the  kinetic  energy  T  is  therefore  a  homogeneous  quad- 
ratic in  q^  <72,  •  •  -,  qn,  the  generalized  component  velocities, 
Lagrange's  equations  can  be  replaced  by  a  set  known  as  the 
Hamiltonian  equations. 

The  Lagrangian  expression  for  the  kinetic  energy  we  shall 
now  represent  by  T^. 

8T-  dT- 

Let  p.  =  -—2  »   pa  =  -~  >  etc.   be  the  generalized  component 

%i  fya 

momenta.  Then  p^  P?  •  •  •  are  homogeneous  of  the  first  degree 
in  q^  ?2,  •  •  •.  Express  q^  £,,•••  in  terms  of  p^  p2,  .  .  .,  qlt  q2,  .  .  ., 
noting  that  they  are  homogeneous  of  the  first  degree  in  terms 
of  p^  pz,  •  •  •,  and  substitute  these  values  for  them  in  T#  which 
will  thus  become  an  explicit  function  of  the  momenta  and  the 
coordinates,  homogeneous  of  the  second  degree  in  terms  of  the 
former.  This  function  is  called  the  Hamiltonian  expression  for 
the  kinetic  energy,  and  we  shall  represent  it  by  Tp.  Of  course 

T,^Tp.  (1) 

By  Euler's  Theorem, 


therefore  2  T.  =  2  Tp  =  p^  +  pjh  +.-.,.  (2) 

dT  dT 

Let  us  try  to  get  •-—  and  —£  indirectly. 

fyi  fy 

38 


CHAP.  II]         THE  HAMILTONIAN  EQUATIONS  39 

dTp      dTq      STjdfr      dT^dqz 
r  rom  (1),        — -  =  — —  -\ — ^  -**  +  7^  -*=  -)-..., 

dTv      dl 
or  — E  =  — 

But  from  (2), 

2^=^^ 
Subtracting  (3)  from  (4),  we  get 

P  __  9  /'^-v'\ 

Again,  we  have  from  (1), 

dTp      dTj  dq^      dTq  dqz 

cT          dq,          dq2 

or  - —  =  p  -^  -\-p  -^  +  •  •  •.  (o) 


From  (2),      2       '  =  «  -f»,     l 

dPi  dPi  i 

Subtracting  (6)  from  (7),  we  get 

dT 

grfc  CD 

The  Lagrangian  equation 

dcT,      dT,_ 

TJ.  ~^~         "o         —  "* 

at  cq-k       cqk 

dT 
becomes  pk  +  --  =  Qk.  (9) 

d(lk 

?jT 
We  have  also  qk  =  -^.  (10) 

ty* 

The  equations  of  which  (9)  and  (10)  are  the  type  are  known 
as  the  Hamiltonian  equations  of  motion.  The  so-called  canonical 
form  of  the  Hamiltonian  equations  is  somewhat  more  compact 
and  will  be  given  later,  in  Chapter  IV. 


40  THE  HAMILTOXIAN  EQUATIONS  [Am.  15 

The  2n  equations  of  which  (9)  and  (10)  are  the  type  form 
a  system  of  2  n  simultaneous  differential  equations  of  the  first 
order,  connecting  the  n  coordinates  q^  qz,  •  •  «,  <?„,  and  the  n  com- 
ponent momenta  p^  pz,  •  •  •  ,  j»n,  with  the  time,  and  in  order  to 
solve  for  any  one  coordinate  we  must  generally,  as  in  the  case 
of  the  Lagrangian  equations,  form  and  make  use  of  the  whole 
set  of  equations. 

In  concrete  problems  there  is  usually  no  advantage  in  using 
the  Hamiltonian  forms,  but  in  many  theoretical  investigations 
they  are  of  importance.  It  may  be  noted  that  in  the  process  of 
forming  Tp  from  T^,  q^  is  expressed  in  terms  of  the  p's  and  q's, 
and  thus  equation  (10)  is  anticipated. 

To  familiarize  the  student  with  the  actual  working  of  the 
Hamiltonian  forms,  we  shall  apply  them  to  a  few  problems 
which  we  have  solved  already  by  the  Lagrange  process. 

15.  (a)  The  equations  of  motion  in  a  plane  in  terms  of 
polar  coordinates  (v.  Art.  3,  (a)). 

Here  T.  =       r2  +  r^. 


Whence  r  =      ,  (1) 

m 

*=•  (2) 


_ 

cr  mrs 


CHAP.  II]  ILLUSTRATIVE  EXAMPLES  41 

\Pr  -^]fir  =  5«r,  (3) 

L         mr*J 

^S</>  =  fcrty.  (4) 

Our  Hamiltonian  equations  are 


<«> 


ft  =  >•*•  (8) 

If  we  eliminate  pr  and  jt)^,  we  get 


our  familiar  equations. 

(5)  Motion  of  a  bead  on  a  horizontal  circular  wire  (v.  Art.  5,  («)). 

Here  T.  =      a2^2. 


r  = 


ma 


42  THE  HAMILTONIAN  EQUATIONS  [ART.  15 

Integrating,         __+--=£=_ 


maV 
m*aV 

Pa  = 


Vkt  + 


A  _  m 


ak  Vkt  +  m 

a      m  ,      rrr1  .,  m  ,       P1       &Fif| 

0  =  -7  log  [F«  +  m]  +  (7  =  —  log   1  H . 

ak  ak  m  \ 

(c)  The  tractrix  problem  (v.  Art.  6,  (a)). 
Here  T.  =  ^  [i2  +  a2^2  -  2  a  cos  (9xi 


^  =  m  [x  —  a  cos  00], 
PQ  =  m \o?Q  —  <z  cos  Ox\ . 

Whence          x  =  —          —  [opx  +  cos  Ope\  (1) 

ma  sm  a 


We  get  px  =  Rsm  6,  (3) 


We  have  the  condition 

a:  =  w*.  (5) 

With  (5),  /)„  =  *»[»  — a  cos  ^P], 

—  n  cos  0]. 
Substituting  in  (4),  we  get 

j9fl  —  mna  sin  #$  =  0, 
or  maz0  =  0. 


CHAP.  II]  ILLUSTEATIVE  EXAMPLES  43 

Whence  &=C  =  -. 

a 

px  =  mn(\  —  cos  0). 

"  Witt 

p  =  mn  sin  66  — sin  6  —  R  sin  6. 

a 

mn2 

K  = 5 

a 

as  in  Art.  6. 

(d)  The  problem  of  the  two  particles  and  the  table  with  a 
hole  in  it  (v.  Art.  8,  (<f)). 

ftfV\  , 

Here  T.  =  ^  [2  f2  +  (a  -  z)2^]. 

px  =  2  '/wi, 


Weget 


and  pe  =  0. 

p9=  C  =  ma 
ma3 


Whence  2a?  + 

a 

if  2;  is  small,  as  in  Art.  8,  (c?). 


44  THE  HAMILTONIAN  EQUATIONS  [ART.  16 

(0)  The  gyroscope  (v.  Art.  10,  (<?)). 

T.  =  l  [A  (6*  +  sin2  0^2)  +  C(jr  cos  6  +  <£)2]. 

- 

cos  0  +  </>), 


O  /TT 

=  —?  =  ^4  sin2  0ijr  +  6r  cos  ^(-f  cos  0 


2, 

" 


We  get  ^  =  0,  (1) 

^  =  0,  (2) 


cos 


olll 


,5  --  _^  fCC^a2  +  i2)  cos  e  -  CLa  (1  +  cos2  0)1  =  mga  sin  0, 

.4  sin3  0 

.v      ri/  —  (7a  cos  0)  (L  cos  0  —  Co) 

or          ^40  =  ^  -  -  ^-  +  ™#asm0.  (4) 

^4  sin3  0 


as  in  Art.  10, 


T/r  COS  0  +  </)  =  «, 

sin2  0-^  +  Ca  cos  0  =  £, 


16.  The  last  two  problems  have  a  peculiarity  that  deserves 
closer  examination.  Let  us  consider  Art.  15,  (<?).  The  kinetic 
energy  hi  the  Lagrangian  form  T^,  and  therefore*  in  the  Ham- 
iltonian  form  Tp,  fails  to  contain  the  coordinates  </>  and  i^-. 
Moreover,  when  either  of  these  coordinates  is  varied,  the 

r\  T  %  T1  • 

*  Since  — -  = (v.  Art.  14),  it  follows  that  if  a  coordinate  is  missing  in 

dqk  dqk 

T$,  it  is  missing  also  in  Tp. 


CHAP.  II]  IGXORABLE  COORDINATES  45 

impressed  forces  do  no  work.  Hence  two  of  our  Hamiltonian 
equations  assume  the  very  simple  forms 

h  =  o,      p*  =  o, 

which  give  immediately 

p^  =  Ca,  a  constant, 
and  p^  =  L,  a  constant. 

These  enable  us  to  eliminate  p$  and  p^  from  a  third  Hamilto- 
nian  equation  (Art.  15,  (0),  (3)),  which  then  contains  only  the 
third  coordinate  6  and  its  corresponding  momentum  p6. 

This  same  result  might  have  been  obtained  just  as  well  by 
replacing  p^  and  p^  in  Tp  by  their  constant  values  and  then 
forming  the  Hamiltonian  equations  for  6  in  the  regular  way. 
So  that  if  we  are  interested  in  0  only,  and  Tp  has  once  been 
formed  and  simplified  by  the  substitution  of  constants  for  p^ 
and  p^,  the  coordinates  </>  and  ^r  need  play  no  further  part  in 
the  solution.  Should  we  care  to  get  the  values  of  these  ignored 
coordinates,  they  can  be  found  from  the  equations  p^  =  €a, 
p^  =  L,  by  the  aid  of  the  value  of  6  previously  determined. 

In  Art.  15,  (d),  since  pe  =  Q  and pe  =  ma vag,  the  substitution 
of  this  value  for  p&  in  Tp  enables  us  to  solve  the  problem  so 
far  as  x  is  concerned  without  paying  further  attention  to  6. 

To  generalize,  it  is  easily  seen  that  if  the  Lagrangian  form, 
and  therefore  the  Hamiltonian  form,  of  the  kinetic  energy  fails 
to  contain  some  of  the  coordinates  of  a  moving  system,*  and 
if  the  impressed  forces  are  such  that  when  any  one  of  these 
coordinates  is  varied  no  work  is  done,  the  momenta  p^  p2, 
corresponding  to  these  coordinates  are  constant ;  and  that  after 
substituting  these  constants  for  the  momenta  in  question  in  the 
Hamiltonian  form  of  the  kinetic  energy,  the  coordinates  corre- 
sponding to  them  may  be  ignored  in  forming  and  in  solving 
the  Hamiltonian  equations  for  the  remaining  coordinates. 

*  Coordinates  that  do  not  appear  in  the  expression  for  the  kinetic  energy 
of  a  moving  system  are  often  called  cyclic  coordinates. 


46  MODIFIED  LAGRANGIAN  EXPRESSION     [ART.  17 

Unfortunately  the  ignored  coordinates  have  to  be  used  in 
forming  T^  the  Lagrangian  form  of  the  kinetic  energy,  and  in 
deducing  from  it  Tp,  the  Hamiltonian  form  of  the  energy. 

Not  infrequently  this  preliminary  labor  may  be  abridged 
considerably  by  using  a  modified  form  of  the  Lagrangian 
expression  for  the  kinetic  energy  of  the  system,  as  we  shall 
proceed  to  show. 

17.  Routh's  Modified  Form  of  the  Lagrangian  Expression  for 
the  Kinetic  Energy  of  a  Moving  System.  In  forming  the 
Hamiltonian  equations  of  motion  (v.  Art.  14)  we  first  changed 
the  form  of  T^  by  replacing  all  the  generalized  velocities 
<?i'  ?2'  *  '  '  by  their  values  in  terms  of  the  coordinates  q^  <?2, 

c  T- 
•  •  •  and  the  generalized  momenta  p^  p^  •  •  • ,  where  pt  =  —-^ , 

f)  T.  ^i 

p  =—i,  etc. 

Let  us  now  try  the  experiment  of  replacing  in  T^  one  only 
of  the  velocities  q^  by  its  value  in  terms  of  the  corresponding 
momentum  p^  the  coordinates  q^  q^  •  •  •,  and  the  remaining 
velocities  £2,  qs, 

Call  TJ  thus  changed  in  form,  Tp.    Of  course 

T_  =  T-,     and     a. 


We  have 


?2i 


Transposing,  =         -ft       =       [2i,  -M]. 

A      •  d-Fpi      3Tj  do,  cq,        c 

Agam,  —  —  l-  =  —  *  -ii  =  p  -^  =  —  i»,«,  j  —  q. 

Pl 


O 

Hence  -  £  =  — 


is  called  the  Lagrangian  expression  for  the  kinetic  energy  modified 
for  the  coordinate  q^ 


CHAP.  II]     MODIFIED  LAGRANGIAN  EXPRESSION  47 

Our  Lagrangian  equation 

d_W*_W*  =  Q 

dt  dq^       dq1         1 

becomes  p.  — —^  =  Q,.  (V) 

%x 

We  have  also  a.  = •  (2) 

§p, 

It  is  noteworthy  that  (1)  and  (2)  differ  from  the  Hamil- 
tonian  equations  for  ql  only  in  that  the  negative  of  the 
modified  expression  Mqi  appears  in  place  of  the  Hamiltonian 
expression  Tp. 

Let  us  go  on  to  the  other  coordinates. 


whence 


whence  .k_ft ;          [^  -^ 

^2  %2  ^2          %, 

The  Lagrangian  equation  for  <?2  is  therefore 
ddMqi      8 

TJ.~*~- 

dt   dq2 

and  differs  from  the  ordinary  form  of  the  Lagrangian  equation 
only  in  that  T^  is  replaced  by  the  modified  expression  Mqi. 

In  forming  the  modified  expression  it  must  be  noted  that  qt 
must  be  replaced  by  its  value  in  terms  of  p^  <?2,  qs,  •  •  • ,  q^ 
q2,  •  •  • ,  not  only  in  T.  but  in  the  term  p1q1  as  well. 

An  advantage  of  the  modified  form  is  that  when  it  has  once 
been  formed  we  can  get  by  its  aid  Hamiltonian  equations  for 
one  coordinate  and  Lagrangian  equations  for  the  others. 


48  MODIFIED  LAGRANGIAN  EXPRESSION     [ART.  18 

The  reasoning  just  given  can  be  extended  easily  to  the  case 
where  we  wish  Hamiltonian  equations  for  more  than  one  coordi- 
nate and  Lagrangian  equations  for  the  rest. 

The  results  may  be  formulated  as  follows:  Let  ^,,p.,---,Pp  be 
the  form  assumed  by  T^  when  q^  q2,  •  •  •  ,  qr,  are  replaced  by 
their  values  in  terms  of  p^  pa,  -  •  .,  pr,  jr+1,  jp+a,  •  •  •,  <jn,  q^ 
q2,  •  •  .  ,  qn.  Then,  if 


we  have  equations  of  the  type 

g^,,.^-..^  _ 
** 


if  A<r  +  l; 

and 

^        "*  ^9* 

if  A:  >  r. 

18.  If  we  modify  the  Lagrangian  expression  for  the  kinetic 
energy  for  all  the  coordinates, 

^.-.fc  =  TP  -PA  -PA  -----  P*fo 
and  we  get  Hamiltonian  equations  of  the  form 


.  ;,..  _.          - 

for  all  the  coordinates,  and  as  we  have  nowhere  assumed  in  our 
reasoning  that  T^  is  a  homogeneous  quadratic  in  the  general- 
ized velocities,  we  can  use  these  equations  safely  when  the 
geometrical  equations  contain  the  time  explicitly  (v.  Art.  7). 
If  the  geometrical  equations  do  not  involve  the  time,  in  which 
case  T^  is  a  homogeneous  quadratic  in  q^  qz,  •  •  •  , 

2  T4  =PA 


CHAP.  II]                ILLUSTRATIVE  EXAMPLE  49 

by  Euler's  Theorem  ;  and  l/9l,..,7n  =  Tp-2Tp  =  -Tp;  and  (1) 
and  (2)  assume  the  familiar  forms 

A  +        -<?«  («> 


It  is  important  to  note  that  the  modified  Lagrangian  expression 
Mgv  gff  ...,qr  is  not  usually  the  kinetic  energy  of  the  system,  although, 
as  we  shall  see  later,  in  some  special  problems  it  reduces  to  the 
kinetic  energy.  As  we  have  just  seen,  when  the  time  does 
not  enter  the  geometrical  equations,  the  completely  modified 
Lagrangian  expression  (that  is,  the  Lagrangian  expression  modi- 
fied for  all  the  coordinates)  is  the  negative  of  the  energy. 

19.  As  an  illustration  of  the  employment  of  the  Hamiltonian 
equations  when  the  geometrical  equations  contain  the  time,  let 
us  take  the  tractrix  problem  of  Art.  6,  («'). 


Here  T  =         *  +  <*     ~^an  cos 


and  is  not  homogeneous  in  6. 


o  /Tn_ 

pd  =  —  r  =  m  \_a?u  —  an  cos 


and 


a  m 

0  n 


9  2    •     /)          a    ,   n    •     a 

-—2  =  mn*  sm  0  cos  0  4-  -  sm  6pd  ; 
cO  a 

n 
and  we  have     pe  —  mn2  sin  6  cos  0  --  sin  Ope  =  0  ;  (2) 

(A> 

and  (1)  and  (2)  are  our  required  Hamiltonian  equations.    Let 
us  solve  them. 


50  MODIFIED  LAGRAXGIAN  EXPRESSION     [ART.  20 

From  (1),  p0  =  ma?6  —  mna  cos  0, 

whence  p9  =  ma20  +  mna  sin  66. 

Substituting  in  (2), 

ma20  -f-  mna  sin  66  —  •  mn2  sin  6  cos  6  —  mna  sin  0$  +  mnz  sin  0  cos  6  =  0, 
or  0=0, 

which  agrees  with  the  result  of  Art.  6,  (a'). 

EXAMPLES 

1.  Work  Art.  6,  (5'),  by  the  Hamiltonian  method. 

2.  Work  Exs.  1  and  2,  Art.  6,  by  the  Hamiltonian  method. 

20.  (a)  As  an  example  of  the  employment  of  the  modified 
form,  we  shall  take  the  tractrix  problem  of  Art.  6  and  modify 
for  the  coordinate  x. 

We  have  (v.  Art.  6) 

T«  =  ?  D*2  +  a^2  -  2  a  cos  6x6]. 
A 

px  =  m  [x  —  a  cos  66~\, 

x  =  &  +  a  cos  66, 
m 


Mx  =  TPt  -pjc  =  +  a2  sin2  0<?2    -       -  a  cos 


mm 


o  1*- 

—  ^  =  ma2  sin2  66  —  a  cos 


=  ma2  sin  6  cos  #0'2  +  a  sin  06. 


CHAP.  II]  ILLUSTRATIVE  EXAMPLES  51 

We  have  for  x  the  Hamiltonian  equations 

px  =  R  sin  0,  (1) 

x  =  £*  +  a  cos  6$,  (2) 

m 

and  for  6  the  Lagrangian  equation 

ma2  [sin2  00  +  sin  0  cos  00s]  -  a  cos  0px  =  0.  (3) 

Of  course  (1),  (2),  and  (3)  must  be  solved  as  simultaneous 
equations,  and  we  can  simplify  by  the  aid  of  the  condition 

x  =  nt.  (4) 

Solving,  we  get  0  =  0, 

R^1^.     (V.  Art.  6  and  Art.  15,  (c)) 


(5)  As  a  second  example  we  shall  take  the  problem  of  the 
two  particles  and  the  wble  with  a  hole  in  it  (v.  Art.  8, 
and  modify  for  0. 

fyfl  "L.      , 

We  have  T.  =  ^  [2  z2  +  (a  -  x)  6>2]. 


O  rji 

pe  =  ^-r  =  m  (a 
c6 


whence  (9=  —  ^-^-  (1) 

m(a  —  xy 


_m   f)  -2 
- 


(a  — 


Our  Hamiltonian  equations  are  (1)  and 

^  =  0.  (2) 


52  MODIFIED  LAGKANGIAN  EXPKESSIOX     [ART.  20 

Our  Lagrangian  equation  is 


By  (2),  p6=C= 

Q 

whence  (3)  becomes  2  x  -\  ---  ^—  -  =  y,  (4) 

(a  —  x) 

as  in  Art.  8,  (<f). 

(c)  As  a  third  example  we  shall  take  the  wedge  and  sphere 
problem  of  Ex.  3,  Art.  9,  and  modify  for  x. 

We  have 

1  ,  .>t      -in  [a2  +  Jt?  ., 

T.  =  -(Jf+w)2r2  +  -|^—  J—  / 

dT- 

px  =  —  r  =  (M  +  m)x  —  my  cos  a. 


_pl  —  m^if-  cos2  a      m  a2  +  k2  .2 
"~  2"      a2     ^  ' 


_pl~  t^y1  cos2  a      TW  a2  +  F  .2      p^  4-  ra/^y  cos  a 

2"      a2      ^  " 


Our  Hamiltonian  equations  are  (1)  and 


Our  Lagrangian  equation  is 

[a2  -(-  &2      ?w2  cos2  a]  ..     mpr  cos  a: 
m—  \v  --  —  =  mq  sin  a.          (&\ 

a2  M+m\y        M+m 

By  (2),  P.-^.-~0;  (4) 

,Q,  ,  Ta2  +  F      mcos2a"|..          .  ,K, 

whence  (3)  becomes         —  -  ----  --  ^  =  g  sm  a,  (5) 

C£  J\L  ~\~  HH\j  \ 

as  in  Art.  9,  Ex.  3. 


CHAP.  II]  ILLUSTRATIVE  EXAMPLES  53 

(eT)  As  a  fourth  example  we  shall  take  the  flexible-parallelo- 
gram problem  (v.  Art.  9,  (£>))  and  modify  for  </>. 

We  have  T.  =  ^  [2  F<£2  +  a202]. 

M 


_ 

- 


pi 


Our  Hamiltonian  equations  are  (1)  and 

A  =  °- 
Our  Lagrangian  equation  is 

ma26  =  —  mga  sin  0, 


or 


EXAMPLES 
1.  Take  the  dumb-bell  problem  of  Art.  8,  (c),  and  modify  for  0. 

Am.    Tpf.  =  \ 


mx  =  0. 
my  =  0. 


54  IGNORATION  OF  COORDINATES  [ART.  21 

2.  Take  the  dumb-bell  problem  of  Art.  8,  (<?),  and  modify  for 
x  and  y. 


=. 

3.  Take  the  gyroscope  problem,  Art.  10,  (e),  and  modify  for 
.  ^  .      (^-^c 


n 
J' 


Vi     -0 

^~ 


coS       _         sin 

21.  We  proceed  to  comment  on  the  problems  of  the  pre- 
ceding section. 

(a)  No  one  of  them  involves  the  time  explicitly  in  the 
geometrical  equations,  and  therefore  the  kinetic  energy  T^  in 
all  of  them  is  a  homogeneous  quadratic  in  the  generalized 
velocities. 

(5)  The  momenta,  therefore,  are  homogeneous  of  the  first 
degree  in  the  velocities,  and  consequently  the  eliminated  veloci- 
ties are  homogeneous  of  the  first  degree  in  the  corresponding 
momenta  and  the  remaining  velocities,  and  the  energy  TPvPv... 
and  the  modified  function  Mq^  Qa,...  are  homogeneous  quadratics 
in  the  introduced  momenta  and  the  velocities  not  eliminated. 


CHAP.  II]          IGNOEATION  OF  COORDINATES  55 

(c)  In  all  the  problems  the  coordinates  for  which  we  have 
modified  the  Lagrangian  expression  for  the  kinetic  energy  are 
cyclic.  In  all  of  them  except  the  first  no  work  is  done  when 
any  one  of  the  coordinates  in  question  is  varied.  Consequently 
one  of  the  Hamiltonian  equations  for  that  coordinate  is  of  the 
form  pk  =  0,  and  the  momentum  pk  =  ck,  where  ck  is  a  constant. 
Therefore  it  is  easy  to  express  the  energy  TPvPv...  and  the 
modified  expression  Mqv9y...  in  terms  of  the  remaining  coordi- 
nates, the  corresponding  velocities,  and  the  constants  c^  c^  •  •  •, 
and  when  so  expressed  they  are  quadratics  in  the  velocities  but. 
not  necessarily  homogeneous  quadratics.  When  the  modified 
function  has  been  so  expressed,  it  may  be  used  in  forming  the 
Lagrangian  equations  for  the  remaining  coordinates  precisely 
as  the  Lagrangian  expression  for  the  kinetic  energy  is  used, 
and  the  coordinates  that  have  been  eliminated  may  be  ignored 
in  the  rest  of  the  work  of  solving  the  problem  unless  we  are 
interested  in  their  values  (v.  Art.  16). 

(c?)  To  generalize :  If  some  of  the  coordinates  of  a  moving  sys- 
tem are  cyclic,  and  if  the  impressed  forces  are  such  that  when  any 
one  of  these  coordinates  is  varied  no  work  is  done,  the  momenta 
corresponding  to  these  coordinates  are  constant  throughout  the 
motion.  The  substitution  of  these  constants  for  the  momenta  in 
the  Lagrangian  expression  for  the  kinetic  energy  modified  for  the 
coordinates  in  question  will  reduce  it  to  an  explicit  function  of 
the  remaining  coordinates,  the  corresponding  velocities,  and  the 
constants  substituted,  which  will  be  a  quadratic  in  the  velocities 
but  not  necessarily  a  homogeneous  quadratic. 

When  the  modified  function  has  been  so  expressed,  it  may 
be  used  in  forming  the  Lagrangian  equations  for  the  remaining 
coordinates  precisely  as  the  Lagrangian  expression  for  the 
kinetic  energy  is  used,  and  the  coordinates  that  have  been 
eliminated  may  be  ignored  in  the  rest  of  the  work  of  solving 
the  problem  (v.  Art.  16). 

In  the  important  case  where  the  system  starts  from  rest,  the 
constant  momenta  corresponding  to  the  ignorable  coordinates 


56  IGNORATION  OF  COORDINATES  [ART.  22 

being  zero  at  the  start  are  zero  throughout  the  motion,  and  the 
modified  expression  is  identical  with  the  Lagrangian  expression 
for  the  kinetic  energy,  which  therefore  is  a  function  of  the 
remaining  coordinates  and  the  corresponding  velocities  and  is 
a  homogeneous  quadratic  in  the  velocities  (v.  Art.  24,  («)). 

(e)  The  fact  that  the  Lagrangian  expression  for  the  kinetic 
energy  modified  for  ignorable  coordinates  is  expressible  in  terms 
of  the  remaining  coordinates  and  the  corresponding  velocities 
and  is  a  quadratic  in  terms  of  those  velocities  is  often  of  great 
importance,  as  we  shall  see  later. 

22.  Let  us  take  the  gyroscope  problem  of  Art.  10,  (<;),  and 
Art.  20,  Ex.  3,  and  work  it  from  the  start,  ignoring  the  cyclic 
coordinates  <j>  and  T|T. 

We  have  T.  =  l  [^  (<92  +  sin2  0^2)  +  C(jr  cos0  +  <£)2],  and 
therefore  <f>  and  -«/r  are  cyclic  coordinates.  Moreover,  no  work 
is  done  when  <£>  is  varied  nor  when  i/r  is  varied,  so  that  <f>  and 
ty  are  ignorable. 

Pi  =  0,     and    pi  =  0,     so  that    p^  =  clt     and    p^  =  c2. 

Q  m 

We  have          p^  =  —  ^  =  C(f  cos  0  +  <£)  =  ef 


ciT- 
=  —2  =  A  sin2       +  C'ea*0(r  cos 


:        C,        (C9  —  C,  COS  0)  COS  6 

whence  <£=-i-.^  -  ^         '       - 

(7  A  sur  ^ 


-  g    COS 


.4  sin2 


CHAP.  II]     TOTAL  IGNORATION  OF  COORDINATES           57 
Forming  the  Lagrangian  equation  for  6  in  the  usual  way,  we 

have        .«      (jet  +  c|)  cos  0  -  c.c,  (1  +  cos2  6} 

Ad  —  -^  -  —  -  .   -1.  ^  --  L  =  mqa  sin  6, 
A  sin3  6 

A  a   ,    (ci  ~  c-2  cos  0)  (C2  —  c\  cos  #)  •     a 

or  Ad  +  -^  -  -  -  1  y.  V,  —  -  =  m9a  sin  0, 

-4  sin  0 

which  is  identical  with  (7),  Art.  10,  (<?). 


EXAMPLE 

Work  the  problem  of  Art.  20,  (6),  ignoring  the  coordinate  0. 

23.  The  problem  of  Art.  20,  (d),  has  an  interesting  peculiarity. 
Both  coordinates  are  cyclic,  and  there  is  no  term  in  the  kinetic 
energy  that  is  linear  in  <£,  the  velocity  corresponding  to  the 
ignorable  coordinate  <f>.    Consequently  the  constant  momentum 
p^  is  a  constant  multiple  of  </>,  which  is  therefore  itself  a  constant. 
This  is  true  also  of  p^>  and  of  ?n&2</>2,  the  term  in  the  energy 
which  involves  <£>.    Tp<ft  and  J/0  must  then  differ  from  mcfQ2  by  con- 
stants, and  as  only  the  derivatives  of  M^  are  used  in  forming  the 
Lagrangian  equation  for  0,  we  have  merely  to  disregard  the  term 
mk~$2  in  the  energy  T^  and  use  what  is  left  of  T^  instead  of  M^. 

In  cases  like  this  we  are  able  practically  to  ignore  the  con- 
tribution to  the  energy  made  by  the  ignored  coordinate  as  well 
as  the  coordinate  itself,  and  of  course  we  can  conclude  that  the 
motion  we  may  thus  disregard  has  no  effect  on  the  motion  we 
are  studying,  but  that  the  two  can  go  on  together  without 
interference. 

EXAMPLE 

Examine  Exs.  1  and  2,  Art.  20,  from  the  point  of  view  of 
the  present  article. 

24.  (a)  The  wedge  and  sphere  problem  of  Art.  20,  (e),  belongs 
to  a  very  important  class.  Both  x  and  y  are  cyclic  coordinates,  and 
x  is  ignorable. 

C.  ip  f 

px  =  -^-r  =  (  M  +  m)x  —  my  cos  a,         pr  =  0. 


58  IGNOKATIOX  OF  COORDINATES  [AKT.  24 

The  momentum  px  is  constant,  and  as  it  is  initially  zero 
(since  the  system  starts  from  rest)  it  is  zero  throughout  the 
motion,  as  is  pxx.  Consequently  the  kinetic  energy  Tp  and  the 
modified  expression  Mx  are  identical,  and  as  they  are  both 
homogeneous  quadratics  in  y  and  px  and  do  not  contain  y,  they 
reduce  to  the  form  Zy2,  where  L  is  a  constant. 

Therefore  our  Lagrangian  equation  for  y  is  Ly  =  mg  sin  a, 
and  the  sphere  rolls  down  the  wedge  with  constant  acceleration. 
If  we  care  only  for  the  motion  of  the  sphere  on  the  wedge,  we 
may  then  ignore  x  completely  and  yet  know  enough  of  the 
form  of  Mx  to  get  valuable  information  as  to  the  required 
motion.  Of  course  we  know  that  the  energy  of  the  whole 
system  can  be  expressed  in  terms  of  y,  and  if  we  are  able  by 
any  means  to  so  express  it,  we  can  solve  completely  for  y  with- 
out using  the  ignored  coordinate  x  at  any  stage  of  the  process. 

(5)  As  a  striking  example  of  this  complete  ignoration  of 
coordinates,  and  of  dealing  with  a  moving  system  having  an 
infinite  number  of  degrees  of  freedom,  let  us  take  the  motion 
of  a  homogeneous  sphere  under  gravity  in  an  infinite  incom- 
pressible liquid,  both  sphere  and  liquid  being  initially  at  rest. 

From  considerations  of  symmetry,  the  position  of  the  sphere 
can  be  fixed  by  giving  a  single  coordinate  x,  the  distance  of 
the  center  of  the  sphere  below  a  fixed  level,  and  x  is  clearly  a 
cyclic  coordinate. 

The  positions  of  the  particles  of  the  liquid  can  be  given  in 
terms  of  x  and  a  sufficiently  large  number  (practically  infinite) 
of  coordinates  q^  q2,  •  •  •,  in  a  great  variety  of  ways.  Assume 
that  a  set  has  been  chosen  such  that  all  the  q's  are  cyclic.* 
Then,  since  gravity  does  no  work  unless  the  position  of  the' 
sphere  is  varied,  the  ^'s  are  all  ignorable.  That  is,  for  every  one 
of  them  pk  =  0,  and  the  momentum  pk  =  ck,  and  since  the  system 
starts  from  rest  the  initial  value  of  pk  is  zero,  and  therefore 
ck  —  0.  Mqv  9j. . . . ,  the  energy  of  the  system  modified  for  all  the  <?'s, 

*  That  this  is  possible  will  be  shown  later,  in  connection  with  the  treatment 
of  Impulsive  Forces  (v.  Chap.  Ill,  Art.  36). 


CHAP.  II]  SUMMAEY  59 

is  then  identical  with  the  energy  TPvP^...  and  must  be  expressi- 
ble in  terms  of  the  remaining  coordinate  x  and  the  corresponding 
velocity  x  (v.  Art.  21,  (c?))  ;  and  as  x  is  cyclic  the  energy  of  the 
system  will  then  be  of  the  form  Lx\  where  L  is  a  constant. 
Forming  the  Lagrangian  equation  for  xt  we  have 

L'x  =  mg, 

and  we  learn  that  the  sphere  will  descend  with  constant 
acceleration. 

Of  course  this  brief  solution  is  incomplete,  as  it  gives  no  infor- 
mation as  to  the  motion  of  the  particles  of  the  liquid,  and  since 
we  do  not  know  the  value  of  L,  we  do  not  learn  the  magnitude 
of  the  acceleration.  Still  the  solution  is  interesting  and  valuable. 

The  energy  of  the  moving  liquid,  calculated  by  the  aid  of 
hydromechanics,  proves  to  be  ^m'x2,  where  m!  is  one  half  the 
mass  of  the  liquid  displaced  by  the  sphere  (Lamb,  Hydrome- 
chanics, Art.  91,  (3)),  and  therefore  the  energy  of  the  system 
is  1  (m  +  m')  ar2,  and  this  agrees  with  our  result. 

25.  Summary  of  Chapter  II.  The  kinetic  energy  of  a  mov- 
ing system  which  has  n  degrees  of  freedom  can  be  expressed 
in  terms  of  the  n  coordinates  q^  q2,  •  -  .,  qn,  and  the  n  general- 
ized momenta  j>?  P&  •  •  •>  Pn->  and  when  so  expressed  it  is  a 
homogeneous  quadratic  in  the  momenta  if  the  geometrical 
equations  do  .not  involve  the  time  (v.  Art.  14),  and  is  called 
the  Hamiltonian  expression  for  the  kinetic  energy.  If  T  is 
the  Hamiltonian  expression  for  the  kinetic  energy,  and  T^  the 
Lagrangian  expression  for  the  kinetic  energy, 

8TP          cT±  dTp 

— -  = »     and     — *  =  <?„ 

fyk         2ft  9pt 

The  work  done  by  the  effective  forces  in  a  hypothetical 
infinitesimal  displacement  of  the  system,  due  to  an  infinitesi- 
mal change  Sqk  in  a  single  coordinate  qk,  is  \pk  -f 

L          * 
this  be  written  equal  to  Qk8qk,  the  work  done  by  the  actual 


60  THE  HAMILTOXIAX  EQUATIONS  [ART.  25 

forces  in  the  displacement  in  question,  there  will  result  the 
differential  equation  of  the  first  order, 


The  2w  equations  of  which  this  and 


given  above,  are  the  type  are  known  as  the  Hamiltonian  equa- 
tions of  motion  for  the  system. 

If,  in  the  expression  Tk  —p^  ~P2q2  -----  #4,  &»  ja,  •  •  •»  jr» 
are  replaced  by  their  values  in  terms  of  pv  p^  •  •  .,  pr,  qr+l, 
•••»  ?n>  9i»  ?2'  *  '  *'  &»  the  result  Jf9j  .....  qr  is  the  Lagrangian 
expression  for  the  kinetic  energy  modified  for  the  coordinates 
#i>  *  '  *'  9r-  For  the  coordinates  for  wliich  the  expression  has 
been  modified,  that  is,  for  &O+1,  we  have  Hamiltonian 
equations  of  the  type 

.       dM      . 


For  the  remaining  coordinates,  that  is,  for  k  >  r,  we  have 
Lagrangian  equations  of  the  type 

d  QM     dM 

-  =  Qt.  (v.  Art.  17) 

eft  e7^A       (7^ 

Whether  we  work  from  the  Lagrangian  expression  T^  for  the 
kinetic  energy,  or  the  Hamiltonian  expression  Tp,  or  the  modi- 
fied Lagrangian  expression  -/&/,,,...,  we  are  in  general  led  to  a  set 
of  simultaneous  differential  equations  whose  number  depends 
merely  upon  the  number  of  degrees  of  freedom  in  the  moving 
system,  and  such  that  to  solve  one  we  must  form  and  solve  all. 

If,  however,  some  of  the  coordinates  are  cyclic  (v.  Art.  16), 
and  the  impressed  forces  are  such  that  when  any  one  of  them 


CHAP.  II]  SUMMARY  61 

is  varied  no  work  is  done,  the  corresponding  momenta  are  con- 
stant throughout  the  motion,  and  these  coordinates  are  ignor- 
able  in  the  sense  that  if  constants  are  substituted  for  the 
corresponding  momenta  in  Tp  or  Mqr...,  the  Hamiltonian  equa- 
tions for  the  remaining  coordinates  in  the  former  case  (or 
the  Lagrangian  equations  in  the  latter  case)  can  be  "formed  and, 
if  capable  of  solution,  can  be  solved  without  forming  the  equa- 
tions corresponding  to  the  ignored  coordinates. 

If  the  system  starts  from  rest  and  there  are  ignorable  coor- 
dinates, Jf,it...,  the  Lagrangian  expression  modified  for  the 
ignorable  coordinates,  is  identical  with  the  kinetic  energy  of 
the  system;  and  whether  the  system  starts  from  rest  or  not, 
Mgv...  is  a  quadratic,  but  not  necessarily  a  homogeneous  quad- 
ratic, in  the  velocities  corresponding  to  the  coordinates  which 
are  not  ignorable. 


CHAPTER  III 
IMPULSIVE  FORCES 

26.  Virtual  Moments.   If  a  hypothetical  infinitesimal  displace- 
ment is  given  to  a  system,  the   product  of  any  force  by  the 
distance  its  point  of  application  is  moved  in  the  direction  of 
the  force  is  called  the  virtual  moment  of  the  force,  and  the  sum 
of  all  the  virtual  moments  is  called  the  virtual  moment  of  the 
set  of  forces. 

If  the  forces  are  finite  forces,  the  virtual  moment  is  the  vir- 
tual work  ;  that  is,  the  work  which  would  be  done  by  the  forces 
in  the  assumed  displacement.  If  the  forces  are  impulsive  forces, 
the  virtual  moment  is  not  virtual  work  but  has  an  interpreta- 
tion as  virtual  action,  which  we  shall  give  later  when  we  take 
up  what  is  called  the  action  of  a  moving  system. 

27.  For  the  motion  of  a  particle  under  impulsive  .forces  we 
have  the  familiar  equations 


m     -  «  =  z- 


are  called  the  effective  impulsive  forces  on  the  particle  and  are 
mechanically  equivalent  to  the  actual  forces. 

If  the  point  is  given  an  infinitesimal  displacement, 


is  the  virtual  moment  of  the  effective  forces  and  of  course  is 
equal  to  the  virtual  moment  of  the  actual  forces. 


62 


CHAP.  Ill]  COMPONENT  OF  IMPULSE  63 

If  the  generalized  coordinates  of  a  moving  system  acted  on 
by  impulsive  forces  are  ^  ,  q.2,  •  •  •  ,  and  a  displacement  caused 
by  varying  ql  by  Bql  is  given  to  the  system, 

&)  **$  +  (A  -  «0)  V] 


„  ,       „  , 

where  S9]J.  represents  the  virtual  moment  of  the  effective  forces. 

dx      dx 

As  in  Art.  7,  ^  =  ^-' 

^2i     a5i 

rn       f  .  dx          .  8x      m   d     ^ 

I  hereiore  mx  —  =  mx  —  =  —  -  —  (or). 

dch  8^      2  a  ^ 

•--[©,-(1).]"- 
[f],-tf  ].-'- 

(where  fgl^ql  is  the  virtual  moment  of  the  impressed  impulsive 
forces  and  Pq  is  called  the  component  of  impulse  corresponding 
to  q^)  is  our  Lagrangian  equation,  and  of  course  we  have  one 
such  equation  for  every  coordinate  qk. 

Equation  (1)  can  be  written  in  the  equivalent  form 

CKX-Cft).-**-       'Jfj    '       (2) 

28.  Illustrative  Examples,  (a)  A  lamina  of  mass  m  rests  on 
a  smooth  horizontal  table  and  is  acted  on  by  an  impulsive 
force  of  magnitude  P  in  the  plane  of  the  lamina.  Find  the 
initial  motion. 

Let  (2:,  y)  be  the  center  of  gravity  of  the  lamina,  let  6  be  the 
angle  made  with  the  axis  of  X  by  a  perpendicular  to  the  line 
of  action  of  the  force,  and  let  a  be  the  distance  of  the  force 
from  the  center  of  gravity. 


IMPULSIVE  FOKCES  [ART.  28 

*"*]•       O  APP-  A> 


If  the  axes  are  chosen  so  that  a:Q  =  0,  y0  =  0,   and  #0  =  0, 
we  have 

mx  =  0, 

m$  =  P, 

mk*0  =  aP. 
Hence  the  initial  velocities  are 

•      n  •      p  A      ap 

z  =  0,        y  =  -  »        ^  =  -r2' 
m  »wr 

The  velocities  of  a  point  on  the  axis  of  X  at  the  distance  b 
from  the  center  of  gravity  are 


The  point  in  question  will  have  no  initial  velocity  if  b  =  --- 

tfc 

It  follows  that  the  lamina  begins  to  rotate  about  an  instan- 
taneous center  in  the  perpendicular  from  the  center  of  gravity 

a2  _L  fc2 

to  the  line  of  action  of  the  force  at  a  distance  -         -  from 

a 

that  line  and  situated  on  the  same  side  of  the  line  of  the  force 
as  the  center  of  gravity.  This  point  is  called  the  center  of 
percussion. 


CHAP.  Ill]  ILLUSTRATIVE  EXAMPLE  65 

7  (6)  A  wedge  of  angle  a  and  mass  M,  smooth  below  and 
perfectly  rough  above,  rests  on  a  horizontal  plane. 

A  sphere  of  radius  a  and  mass  m  is  rotating  with  angular 
velocity  H  about  a  horizontal  axis  parallel  to  the  edge  of  the 
wedge  and  is  placed  gently  on  the  wedge.  Find  the  initial 
motion  (v.  Art.  9,  Ex.  3). 

Take  as  coordinates  x,  the  distance  of  the  edge  of  the  wedge 
from  a  fixed  axis  parallel  to  it  in  the  horizontal  plane ;  y,  the 
distance  of  the  point  of  contact  of  the  sphere  down  the  wedge ; 
and  #,  the  angle  through  which  the  sphere  has  rotated. 

We  have 

j\£  _i_  yYi   ,      in  ' 

5          2  2 

both  before  and  after  the  sphere  is  set  down.  After  the  sphere 
is  set  down,  y  —  ad  =  0.  Before  it  is  set  down,  x  =  y  =  0  and 
$  =  O.  Since  the  sphere  cannot  slip,  it  exerts  an  impulsive 
force  P  up  the  wedge,  and  an  equal  and  opposite  force  P  is 
exerted  on  it  by  the  wedge  at  the  instant  the  two  bodies  come 
in  contact. 

We  have        px  =  ^-^  =  (M+  m)  x  —  my  cos  a, 

_!?  —  vn  (>ii *r  nnc  f%\ 


Our  equations  are 

(M  +  m)x  —  my  cos  a;  =  0,  (1) 

m(ij  —  x  cos  a)  =  P,  (2) 

rf(<9-H)  =  -aP,  (3) 

and  we  have  also  a0  =  y.  (4) 

From  (2),  (3),  and  (4), 

Ffl  ,^ 

-7-.  (5) 


66  IMPULSIVE  FORCES  [ART.  28 

From  (1)  and  (5), 

\,  ,«2  +  F  ,  "I.      rfn 

(M+  m)  -  -  --  m  cos  a  \x  —  -  11  cos  a, 
L  a  J          a 

and  (M  +  m)  --  -  --  m  cos2  a  \y  =  (  Jf  +  w)  —  fl. 

V/  (c)  In  Art.  8,  (6),  let  the  weight  4  ra  be  jerked  down  with  a 
velocity  v.  Find  the  initial  motion.  Take  the  coordinates  x 
and  y  as  in  Art.  8,  (6),  and  let  P  be  the  magnitude  of  the  jerk. 


=  " 


Our  equations  are      m  (8  x  —  i/}  =  P, 
and  w  (3  ^  —  x)  =  0. 

Whence  23  mx  =  3  P, 

23  »$  =  P. 

3  P 

But  i  =  w  =  —  —  , 

23  w 

and  y  —  ^v- 

(d)  Four  equal  rods  freely  jointed  together  in  the  form  of  a 
square  are  at  rest  on  a  horizontal  table.  A  blow  is  struck  at 
one  corner  in  the  direction  of  one  of  the  sides.  Compare  the 
initial  velocities  of  the  middle  points  of  the  four  rods. 

Let  m  be  the  mass  and  2  a  the  length  of  a  rod.  Take  a  pair 
of  rectangular  axes  in  the  table.  Let  (x,  y)  be  the  center  of  the 
figure  at  any  time  and  6  and  <j>  the  angles  made  by  two  adjacent 
rods  with  the  axis  of  X.  x,  y,  0,  </>,  are  our  generalized  coordinates. 

The  rectangular  coordinates  of  the  four  middle  points   are 

obviously  .     ,N 

(x  —  a  cos  9,  y  —  a  sin  9),  (1) 

(x-\-a  cos  0,    y  +  a  sin  0),  (2) 


CHAP.  Ill]  ILLUSTRATIVE  EXAMPLE  67 

(x  +  a  cos  (/>,  y  -\-  a  sin  <£),  (3) 

(x  —  a  cos  0,    y  —  a  sin  0).  (4) 


We  have 
T*=2 


=  4  my, 


7T 

Let  the  values  of  #,  y,  0,  <f>,  be  0,  0,  0,  —  ,  before  the  blow  is 
struck,  and  let  P  be  the  magnitude  of  the  blow. 
Our  equations  are  4  mx  =  P, 

4  my  =  0, 


and  2w(V 

Whence  y  —  0, 


Let  t'j,  v2,  t'3,  t>4,  be  the  required  velocities  of  the  four  middle 
points.    Then 

•      3a2  +  F     P  5P 

V    =  X  4-  #<!>  ^  •  -       —  •  -  =        --  i 

1  a2  +  /P     47w          8m 

P  2P 


4  m  8m 

•  _     F-a2   P  IP 

Vs~                     ^Ht2!^  ~8m 

P  2P 


and  v'.v  :v-.v  =  5  :  2  :  —  1  :  2. 

1234 


68  IMPULSIVE  FORCES  [ART.  29 

29.  General  Theorems.  Work  done  by  an  Impulse.  If  a  particle 
(x,  y,  z)  initially  at  rest  is  displaced  to  the  position  (.r  -f  &B, 
y  +  Sy,  z  +  £2),  the  displacement  can  be  conceived  of  as  brought 
about  in  the  interval  of  time  St  by  imposing  upon  the  particle 
a  velocity  whose  components  parallel  to  the  axes  are  u^  v^  w^ 
where  Sx  =  ufit,  Sy  =  vfit,  and  8z  =  wfit. 

If  the  particle  is  initially  in  motion  with  a  velocity  whose 
components  are  u,  v,  w,  the  displacement  in  question  could  be 
brought  about  by  imposing  upon  it  an  additional  velocity  whose 
components  are  obviously  ul  —  u,  vl  —  v,  wl  —  w. 

Let  a  moving  system  be  acted  on  by  a  set  of  impulsive 
forces.  Let  m  be  the  mass  of  any  particle  of  the  system  ; 
Pz,  Pyr  Pz,  the  components  of  the  impulsive  force  acting  on  the 
particle  ;  w,  v,  w,  the  components  of  the  velocity  of  the  particle 
before,  and  u^  v^  wl  after,  the  impulsive  forces  have  acted. 

If  any  infinitesimal  displacement  is  given  to  the  system  by 
which  the  coordinates  of  the  particle  are  changed  by  &r,  8yt 
and  82,  we  have  the  virtual  moment  of  the  effective  forces 
equal  to  the  virtual  moment  of  the  actual  forces  (v.  Art.  27); 
that  is, 

l  —  w)  8x  +  (vl  —  v)§y  +  (wl  —  w*)  8z~] 


If  the  velocity  that  would  have  to  be  imposed  upon  the  parti- 
cle w,  were  it  at  rest,  to  bring  about  its  assumed  displacement 
in  the  time  St  has  the  components  w2,  t>2,  w2,  Sx  =  u.28t,  §y  =  vz&t, 
8z  =  wJ8t,  and  the  equation  above  may  be  written 

2m  [(MJ  —  w)  M2  +  (^  —  v)  v2  +  (wl  —  uf)  wj 

=  2[u2Px  +  v2Py  +  waPz-].       (1) 

Interesting  special  cases  of  (1)  are 

^  —  u)  u  +  (vl  —  v)  v  -t-  (wa  —  w)  w~\ 

t;Pf  +  wP.],       (2) 


(3) 


CHAP.  Ill]  THOMSON'S  THEOREM  69 

the  displacement  used  in  (2)  being  what  the  system  would 
have  had  in  the  time  St  had  the  initial  motion  continued,  and 
that  in  (3)  what  it  has  in  the  actual  motion  brought  about  by 
the  impulsive  forces.  If  we  take  half  the  sum  of  (2)  and  (3), 
we  get 

2  1  K  +  ».'  +  <]  -  2  f  [«"  +  "!  +  <i 


<4> 

or,  a  system's  gain  in  kinetic  energy  caused  by  the  action  of 
impulsive  forces  is  the  sum  of  the  terms  obtained  by  multiplying 
every  force  by  half  the  sum  of  the  initial  and  final  velocities  of 
its  point  of  application,  both  being  resolved  in  the  direction  of 
the  force. 

This  sum  is  usually  called  the  work  done  by  the  impulsive 
forces. 

30.  Thomson's  Theorem.  If  our  system  starts  from  rest, 
u  .  =  v  =  w  =  0,  and  formulas  (1)  and  (3),  Art.  29,  reduce  re- 
spectively to 

2m  [u^  +  v^vz  +  w^J  =  2  [u2Px  +  v2Pv  +  w2PJ,        (1) 
and  Zm  [<  +  vf  +  wf]  =  2  [_uj>x  +  vfy  +  WIP,].        (2) 

But  the  first  member  of  (1)  is  identically 


and  subtracting  (2)  from  (1)  we  get 


If  w2,  v2,  M'2,  are  the  components  of  velocity  of  the  particle  m  in 
any  conceivable   motion  of  the   system  which   could  give   the 


70  IMPULSIVE  FORCES  [ART.  31 

points  of  application  of  the  impulsive  forces  the  same  velocities 
that  they  have  in  the  actual  motion,  then  the  second  member  of 
(3)  is  zero  and  we  get  Thomson's  Theorem'. 

If  a  system  at  rest  is  set  in  motion  by  impulsive  forces,  its  kinetic 
energy  is  less  than  in  any  other  motion  where  the  velocities  of  the 
points  of  application  of  the  forces  in  question  are  the  same  as  in  the 
actual  motion,  by  an  amount  equal  to  the  energy  the  system  would 
have  in  the  motion  which,  compounded  with  the  actual  motion,  would 
produce  the  hypothetical  motion. 

31.  Bertrand's  Theorem.  If  Qx,  Qy,  and  Qz  are  the  com- 
ponents of  the  impulsive  force  which  would  have  to  act  on  the 
particle  m  of  the  system  considered  in  Art.  29  to  change  its 
component  velocities  from  u,  v,  w,  to  uz,  v2,  wz,  formula  (3), 
Art.  29,  gives  us 


=  2[uag,  +  ^F  +  frf^].        (1) 
Subtracting  (1)  from  (1)  in  Art.  29,  we  get 


The  first  member  of  (2)  is  (v.  Art.  30)  identically 

2  f  (  w  +  *'  t  <i  -  1>  *  +  "•  +  W22] 

-   [(«,  -  'O'  +  (^  -  tO"  +  K  -  ^)2]>- 

If  the  second  member  of  (2)  is  zero,  as  will  be  the  case  if  the 
^-forces  differ  from  the  P-forces  only  by  the  impulsive  actions 
and  reactions  due  to  the  introduction  of  additional  constraints 
which  have  no  virtual  moment  in  the  hypothetical  motion  into 
the  original  system,  we  have  Bertrand's  Theorem: 

If  a  system  in  motion  is  acted  on  by  impulsive  forces,  the  kinetic 
energy  of  the  subsequent  motion  is  greater  than  it  would  be  if  the 
system  were  subjected  to  any  additional  constraints  and  acted  on 


CHAP.  Ill]  THOMSON'S  THEOREM  71 

by  the  same  impulsive  forces,  by  an  amount  equal  to  the  energy  it 
would  have  in  the  motion  which,  compounded  with  the  first  motion, 
would  give  the  second.* 

32.  By  the  aid  of  Thomson's  Theorem  many  problems  in- 
volving impulsive  forces  can  be  treated  as  simple  questions  in 
maxima  and  minima. 

(a)  If,  for  example,  in  Art.  28,  (a),  instead  of  giving  the 
force  P  \ve  give  the  velocity  v  of  the  foot  of  the  perpendicular 
from  the  center  of  gravity  upon  the  line  of  the  force,  so  that 
y  +  a0  =  v,  then  to  find  the  motion  we  have  only  to  make  the 

77i  • 

energy  T^  =  —  [i2  +  y2  +  fi?62~]  a  minimum. 


t         .  A 
—  -  =  mx  —  0, 

ex 

dT-  Jc2 

—  q-  =  m  [y  --  :  (v  —  w)l  =  0. 

cy           Lif  a^        Un 

Whence  x  =  0, 


a  w 

j-t      n* 

2  I,2 

and  these  results  agree  entirely  with  the  results  obtained  in 
Art,  28,  (a). 

*  Gauss's  Principle  of  Least  Constraint :  If  a  constrained  system  is  acted  on 
by  impulsive  forces,  Gauss  takes  as  the  measure  of  the  "constraint"  what  is 
practically  the  kinetic  energy  of  the  motion  which,  combined  with  the  motion 
that  the  system  would  take  if  all  the  constraints  were  removed,  would  give  the 
actual  motion. 

It  follows  easily  from  Bertrand's  Theorem  that  this  "constraint"  is  less 
than  in  any  hypothetical  motion  brought  about  by  introducing  additional  con- 
straining forces  (v.  Routh,  Elementary  Rigid  Dynamics,  §§  391-393). 


72  IMPULSIVE  FOKCES  [ART.  33 

(6)  In  Art.  28,  (c),  since  x  =  v,  the  energy 

^  =  ^[8^  +  3f-2^]. 
To  make  Tq  a  minimum,  we  have 


*"!' 

and  the  problem  is  solved. 

(f)  In  Art.  28,  (cT),  let  t>  :  =  x  +  a<£  be  given. 

Then  T.  =  5J  [4^  -  a<^)2  +  4f  +  2(a2  +  &2)(0'2  + 


- 
-^  =  m[2(a2 


and 


3  a2  +  A:2 

#2  -i-  ft?  2 

3  a2  +  F l>1  ~"  r2  ~  V*  ~  5 


33.  In  using  Thomson's  Theorem  we  may  employ  any  valid 
form  in  the  expression  for  the  energy  communicated  by  the 
impulsive  forces.  For  instance,  in  the  case  of  any  rigid  body, 

T  =  |  [i2  +  f  +  ?  +Aa>l  +£<  +  C<] 

is  permissible  and  is  much  simpler  than  the  corresponding  form 
in  terms  of  Euler's  coordinates. 


CHAP.  Ill]  THOMSON'S  THEOREM  73 

Take,  for  example,  the  following  problem  :  An  elliptic  disk 
is  at  rest.  Suddenly  one  extremity  of  the  major  axis  and  one 
extremity  of  the  minor  axis  are  made  to  move  with  velocities 
U  and  V  perpendicular  to  the  plane  of  the  disk.  Find  the 
motion  of  the  disk. 

Let  us  take  the  major  axis  as  the  axis  of  X  and  the  minor 
axis  as  the  axis  of  Y. 

fffi 

We  have,  then,  T  =  -£  [i?  +  f  +  ?  +  A<o*  +  Ba>*  +  O»88]. 

ti 

By  the  conditions  of  the  problem,  since  the  components  of  the 
velocity  of  the  point  (a,  0,  0)  are  0,  0,  U,  and  those  of  the 
point  (0,  b,  0)  are  0,  0,  V,  we  have 

i  =  0, 

if  +  aa)s  =  0, 
2  —  aeo2  =  U; 
and  x  —  bo)  =  0, 


2  +  &o>a  =  V. 
Hence         T=       z2  +     <T-^)2  + 


- 

mb*  ma2 


since 


=  ^r-     and     B  = 


4  4 

~dz=' 


We  have  also  co  =  —  (5  V  —  U\ 

b  o 

2      6  a 


74  IMPULSIVE  FORCES  [ART.  34 

EXAMPLE 

One  extremity  of  a  side  of  a  square  lamina  is  suddenly  made 
to  move  perpendicular  to  the  plane  of  the  lamina  with  velocity 
V,  while  the  other  extremity  is  made  to  move  in  the  plane  of 
the  lamina  and  perpendicular  to  the  side  with  velocity  V. 

Show  that  the  center  will  move  with  velocity  —  perpendicular 

V    - 
to  the  plane,  and  with  velocity  —  \/2  in  the  plane,  toward  the 

a 

corner  on  which  the  velocity  V  was  impressed. 

34.  If  our  system  does  not  start  from  resty  it  is  often  easy 
to  frame  and  to  solve  a  problem  in  which  the  system  is  initially 
at  rest  and  is  acted  on  by  the  same  impulsive  forces  as  in  the 
actual  problem,  and  .where  consequently  the  resulting  motion, 
compounded  with  the  actual  initial  motion,  will  give  the  actual 
final  motion. 

For  example,  consider  the  following  problem :  A  sphere 
rotating  about  any  axis  is  gently  placed  on  a  perfectly  rough 
horizontal  plane.  Find  the  initial  motion.  Here,  in  the  actual 
case,  the  lowest  point  of  the  sphere  is  immediately  reduced 
to  rest. 

Take  rectangular  axes  of  X  and  Y,  parallel  to  the  plane  and 
through  the  center  of  the  sphere.  Let  flx,  £ly,  flz,  be  the  com- 
ponent angular  velocities  before,  and  a>x,  (0y,  o>z  after,  the  sphere 
is  placed  on  the  plane.  Let  x,  y,  be  the  velocities  of  the  center 
of  the  sphere. 

Then,  in  the  actual  case,  x  —  awy  =  0  and  y  +  aa>x  =  0  are  our 
given  conditions.  Initially  the  velocities  of  the  lowest  point  of 
the  sphere  are  —  afly  and  a£lx.  If  the  sphere  were  at  rest,  the 
impulsive  force  which  in  the  actual  case  destroys  these  velocities 
would  give  to  the  lowest  point  the  negatives  of  these  velocities  ; 
that  is,  a£ly  and  —  a£lx.  Let  us  then  solve  the  following 
auxiliary  problem:  A  sphere  is  at  rest.  Suddenly  the  lowest 
point  is  made  to  move  with  velocities  afly  and  —  «Or,  parallel  to 
a  pair  of  horizontal  axes.  Find  the  initial  motion  of  the  sphere. 


CHAP.  Ill]  ILLUSTRATIVE  EXAMPLE  75 

If  u  and  v  are,  respectively,  the  x  component  and  the  y 
component  of  the  velocity  of  the  center,  and  a^,  &>2,  tu3,  are  the 
angular  velocities, 


T  =      1>2  + 

The  velocities  of  the  lowest  point  are  u  —  ao>2,  v  +  aco^  but 
they  were  given  as  a£ly  and  —  rtO^.. 

Therefore  u  —  «&>,,  =  afl, 


To  make  this  a  minimum,  we  have 


2 


da>3 

cf 

Hence  <o  = —  n_, 

a2  +  ft* 

M»  = 5 — 77,  H  , 

<r  +  /r 


Compounding  these  with  the  initial  angular  velocities  in  the 
actual  problem,  O,,.,  Hy,  flz,  we  get 


O 

' 


These  equations,  together  with  x  —  awy  =  0  and  y  -f  aa)x  —  0, 
completely  solve  the  original  problem. 


76  IMPULSIVE  FOKCES  [ART.  35 

35.  A  Problem  in  Fluid  Motion.  Let  us  now  consider  an  inter- 
esting application  of  the  principles  of  this  chapter  which  was 
made  by  Lord  Kelvin  to  a  problem  in  fluid  motion. 

It  is  shown  in  treatises  on  hydromechanics  that  if  an  incom- 
pressible, frictionless,  homogeneous  liquid,  either  infinite  in 
extent  or  bounded  by  any  finite  closed  surfaces  fixed  or  moving, 
and  with  any  rigid  or  flexible  bodies  immersed  in  it,  is  moving 
under  the  action  of  conservative  forces  (v.  Chap.  IV)  and  has 
ever  been  at  rest,  the  motion  will  be  what  is  called  irrotational, 
That  is,  if  #,  y,  z,  are  the  rectangular  coordinates  of  any  fixed 
point  in  the  space  occupied  by  the  liquid,  there  will  be  a  func- 
tion </>(#,  y,  z)  such  that  if  u,  v,  w,  are  the  components  of  the 
velocity  of  the  liquid  at  the  point  x,  y,  z, 

dd)  cd>  cd> 

u  =  -~')          v  =  — -,          w  =  —  > 
ex  cy  cz 

The  function  </>  is  called  the  velocity-potential  function. 

Since  throughout  the  motion  the  liquid  is  always  supposed 
to  be  an  incompressible  continuum,  u,  v,  w,  must  satisfy  the  equa- 
tion of  continuity  for  an  incompressible  liquid, 

ou      ov      dtv  _  A 

Q          '      O '     "o~  ~   ^  ' 

ox      oy      cz 

and  therefore  <£  satisfies  Laplace's  equation, 
V$     P+     224>_0 

O     2   ~T~    o     1     •     ~^~2    U' 

dx2      dy2      dz2 
and  will  be  uniquely  determined  except  for  an  arbitrary  constant 

term  if  the  value  of  -^,  the  velocity  normal  to  the  surface,  is 
dn 

given  at  every  point  of  the  boundary  of  the  liquid,  however 
irregular  that  boundary.  Therefore  the  actual  motion  at  every 
point  of  the  liquid  at  any  instant  is  uniquely  determined,  if  the 
motion  is  irrotational,  when  the  normal  velocities  at  all  points 
of  the  boundary  are  given. 

We  wish  now  to  prove  that  the  kinetic  energy  of  the  actual 
motion  is  less  than  that  of  any  other  motion,  not  necessarily 


CHAP.  Ill]  PKOBLEM  IN  THIKD  MOTION  77 

irrotational,  consistent  with  the  equation  of  continuity  and  with 
the  actual  normal  velocities  at  the  boundary. 

If  w,  v,  w,  are  the  velocities  at  (x,  y,  z)  in  the  actual  motion, 
let  u  +  a,  v  +  6,  w  +  7,  be  the  velocities  in  the  hypothetical 
motion,  and  let  vn  be  the  actual  normal  velocity  at  any  point 

O  J 

of  the  boundary.    Then  we  have  ^-  =  lu  -f  mv  +  nw  =  #„,  where 

dn 

£,  m,  n,  are  the  direction  cosines  of  the  normal,  and  u,  v,  w,  are 
the  components  of  the  velocity  at  the  point  in  question. 
In  the  hypothetical  motion, 

l(u  +  a)  +  m  (y  +  £)  +  n  (w  +  7)  =  vn 
at  the  same  point  ;  therefore 

la  +  m/3  +  ny  =  0  (1) 

at  every  point  of  the  boundary. 

As  the  hypothetical  velocities  as  well  as  the  actual  velocities 
must  obey  the  law  of  continuity, 

g  (M  +  «)      d  (v  +  ff)      3  (w  4-  7)  _  n 
o  ~r         ^  •          <j  —    ' 

ex  cy  oz 

ca      88      8y      n 
and  therefore  -  +  ^-  +  -^  =  0 

ox      cy      dz 

at  every  point  in  the  bounded  space. 
If  T  is  the  energy  of  the  actual  motion, 

T=  | 

where  />  is  the   density  of  the  liquid,  and  where  the  volume 
integral  is  taken  throughout  the  space  filled  by  the  liquid. 
If  T'  is  the  energy  of  the  hypothetical  motion, 


u  +  «)2  +  (f  +  £)2  +  (w  +  7) 
=  T  +          T[«2  +  £•  +  72] 
+  P  I  I  I  \_au  +  Pv 


78  IMPULSIVE  FORCES  [ART.  35 

By  the  aid  of  Green's  Theorem  we  can  prove  that 

» 

[au  +  (3v  +  7M>]  dx  dy  dz  =  0. 

W 


We  have        I  J  /  ~  —  dxdydz  =  I  U  cos  adS 


(v.  Peirce,  Newtonian  Potential  Function,  p.  92  (143)),  where 
the  volume  integral  is  taken  throughout  any  bounded  space 
and  the  surface  integral  over  the  boundary  of  the  space,  cos  a 
being  the  x  direction  cosine  of  the  normal  to  the  boundary. 


XT 

Now  au  =  a  —  =  —  («$)  —  9  — 

ex      ex  ex 


and         \  (  (  audxdydz  =  \  Ia<f>dS  —  \\\$  — 
In  like  manner, 

CCC/3vdxdydz  =  CmjS^dS-  CCfo  ~  dx  dy  dz, 

and         I  I  I  yw  dx  dy  dz  =  I  nyfalti  —  \\\$^- 
Hence      |  /  /  [au  +  (3v  +  yw~\  dx  dy  dz 

=  C\la  +  m^  +  n^dS-  CCC^-  + 

But  the  surface  integral  vanishes  by  (1),  and  the  volume 
integral  vanishes  by  (2).  Therefore  the  energy  T'  is  greater 
than  the  actual  energy  T. 

It  follows  that  the  irrotational  motion  of  any  frictionless 
incompressible  homogeneous  liquid  under  the  action  of  con- 
servative forces  is  at  every  instant  identical  with  the  motion 
which  would  have  been  suddenly  generated  from  rest  by  a  set  of 
impulsive  forces  applied  at  points  in  the  boundary  of  the  liquid 
and  such  that  they  would  suddenly  give  all  the  points  of  the 
boundary  the  normal  velocities  that  these  points  actually  have 
at  the  instant  in  question  (v.  Thomson's  Theorem  in  Art.  30). 


CHAP.  Ill]  PROBLEM  IX  THIRD  MOTION  79 

36.  If  now  we  have  a  liquid  contained  in  a  material  vessel 
and  containing  immersed  bodies,  a  set  of  generalized  coordinates 
9i>  #2'  '  *  ''  ?«'  can  t>e  chosen,  equal  to  the  number  of  degrees  of 
freedom  of  the  material  system  formed  by  the  vessel  and  the 
immersed  bodies,  and  the  normal  velocity  of  every  point  of  the 
surface  of  the  vessel  and  of  the  surfaces  of  the  immersed  bodies 
can  be  expressed  in  terms  of  the  coordinates  q^  q.2,  •  •  •,  qn,  and 
the  corresponding  generalized  velocities  q^  q2,  •  •  •,  qn. 

\Ve  can  now  choose  other  independent  coordinates  q'v  q'2,  •  •  •, 
practically  infinite  in  number,  which,  together  with  our  coordi- 
nates q^  q2,  •  •  .,  qn,  will  give  the  positions  of  all  the  particles 
of  the  liquid. 

Suppose  the  system  (vessel,  immersed  bodies,  and  liquid)  at 
rest.  Apply  any  set  of  impulsive  forces,  not  greater  in  num- 
ber than  n,  at  points  in  the  surface  of  vessel  and  of  immersed 
bodies  and  consider  the  equations  of  motion.  For  any  of  our 
coordinates  q'k  we  have  the  equation 


where  p'k  is  the  generalized  momentum  corresponding  to  q'tJ 
since  (as  in  varying  q'k  no  one  of  the  coordinates  q^  q2,  -  •  •,  qn, 
is  changed,  and  therefore  no  one  of  the  impulsive  forces  has  its 
point  of  application  moved)  the  virtual  moment  of  the  com- 
ponent impulse  corresponding  to  qk  is  zero. 

As  the  actual  motion  at  every  instant  could  have  been  gen- 
erated suddenly  from  rest  by  such  impulsive  forces  as  we  have 
just  considered,  the  momentum  p'k  is  zero  throughout  the  actual 
motion  ;  and  the  impressed  forces  being  by  hypothesis  conserv- 
ative, and  the  liquid  always  forming  a  continuum,  no  work  is 
done  when  q'k  is  varied.  Consequently,  in  the  Hamiltonian 

equation  p'k  H  --  -  =  Q'k,  pk  =  Q  and  Q'k  =  0,  and  therefore  —  -  =  0. 

*fc  % 

Hence  every  coordinate  q'k  is  cyclic,  and  it  is  also  completely 
ignorable.  The  energy  modified  for  the  coordinates  qk  is  identi- 
cal with  the  energy,  which,  being  free  from  the  coordinates  q't 


80  IMPULSIVE  FOKCES  [ART.  37 

and  the  momenta  p'k,  is  expressible  in  terms  of  the  n  coordinates 
(li>  #2»  '  '  *'  &«'  anc^  *ke  corresPondmg  velocities  q^  qz,  •  •  .,  qn, 
and  is  a  homogeneous  quadratic  in  terms  of  these  velocities 
(v.  Art.  24,  (a)). 

37.  Summary  of  Chapter  III.  In  dealing  with  problems  in 
which  a  moving  system  is  supposed  to  be  acted  on  by  impul- 
sive forces,  we  care  only  for  the  state  of  motion  brought  about 
by  the  forces  in  question,  since  on  the  usual  assumption  that 
there  is  no  change  in  configuration  during  the  action  of  the 
impulsive  forces,  we  are  not  concerned  with  the  values  of  the 
coordinates  but  merely  with  the  values  of  their  time  derivatives. 

The  virtual  moment  (v.  Art.  26)  of  the  effective  impulsive 
forces  in  a  hypothetical  infinitesimal  displacement  of  the  system, 
due  to  an  infinitesimal  change  &qt  in  a  single  coordinate  qk,  is 

' or 

If  either  of  these  is  written  equal  to  PkSqk,  the  virtual  moment 
of  the  actual  impulsive  forces  in  the  displacement  in  question, 
we  have  one  of  the  equivalent  equations 


r?si  _ 
kJ, 


The  n  equations  of  which  either  of  these  is  the  type  are  n 
simultaneous  linear  equations  in  the  n  final  velocities  (5^, 
(^2)1'  •  •  •,  and  as  the  configuration  and  the  initial  state  of 
motion  are  supposed  to  be  given,  no  integration  is  required, 
and  the  problem  becomes  one  in  elementary  algebra. 

A  skillful  use  of  Thomson's  or  of  Bertrand's  Theorem  re- 
duces many  problems  in  motion  under  impulsive  forces  to 
simple  problems  in  maxima  and,  mimima. 


CHAPTER  IV 
CONSERVATIVE  FORCES 

38.  If  X,  Y,  Z,  are  the  components  of  the  forces  acting 
on  a  moving  particle  (coordinates  a:,  y,  z),  the  work,  W,  done 
by  the  forces  while  the  particle  moves  from  a  given  position 
P#  Ov  y0»  zo)'  to  a  second  position  Px,  (ajj,  yx,  Zj),  is  equal  to 


/' 

Jr. 


\_Xdx  +  Ycfy  +  Zdz]  ; 


and  since  every  one  of  the  quantities  X,  F,  and  Z  is  in  the 
general  case  a  function  of  the  three  variables  x,  y,  z,  we  need 
to  know  the  path  followed  by  the  moving  particle,  in  order  to 
find  W.  Let  /(#,  y,  z)  =  0,  <£  (x,  y,  z)  =  0,  be  the  equations  of 
the  path.  We  can  eliminate  z  between  these  equations  and  then 
express  y  explicitly  in  terms  of  x\  we  can  then  eliminate  y 
between  the  same  two  equations  and  express  z  in  terms  of  x; 
and  we  can  substitute  these  values  for  y  and  z  in  X,  which  will 

rxi 

then  be  a  function  of  the  single  variable  x,  and    I     Xdx  can  be 

Jx0 

found  by  a  simple  quadrature.    By  proceeding  in  the  same  way 

r»i  rzi 

with  Y  and  Z  we  can  find   I      Ydy  and   I     Zdz,  and  the 

J*t  J** 

of  these  three  integrals  will  be  the  work  required. 

It  may  happen,  however,  that  Xdx  +  Ydy  +  Zdz  is  what  is 
called  an  exact  differential,  that  is,  that  there  is  a  function 
U  =  </>  (x,  y,  z)  such  that 

!£.,>      du-Y      du-z 

~3~  ~     '  ~^~  ~     '  ~2~  ~~ 

ex  c  cz 


sum 


82  CONSERVATIVE  FORCES  [ART.  38 

Since  the  complete  differential  of  this  function  is 


cU  ,       cU  ,        cU 
—  dx  +  —  dy-\-  —  dz, 
ox  cy  cz 

or  Xdx  +  Ydy  +  Zdz,  we  have 

[  Xdx  +  Ydy  +  Zdz~]  =  <j>  (x,  y,  z)  =  U, 

and         I      \Xdx  +  Ydy  +  Zdz] 


/ 


and  in  obtaining  this  result  we  have  made  no  use  of  the  path 
followed  by  the  moving  particle. 

When  the  forces  are  such  that  the  function  U  =  $  (#,  y,  z) 
exists,  they  are  said  to  be  conservative,  and  U  is  called  the 
force  function. 

We  can  infer,  then,  that  the  work  done  by  conservative 
forces  on  a  particle  moving  by  any  path  from  a  given  initial 
position  to  a  given  final  position  is  independent  of  the  path  and 
is  equal  to  the'  value  of  the  force  function  in  the  final  position 
minus  its  value  in  the  initial  position. 

If  instead  of  a  moving  particle  we  have  a  system  of  particles, 
the  reasoning  given  above  applies. 

Let  (xk,  yj.,  ZA.)  be  any  particle  of  the  system,  and  Xk,  Yk,  Zk, 
be  the  forces  applied  at  the  particle.  Then  the  whole  work,  If, 
done  on  the  system  as  it  moves  from  one  configuration  to 
another  is  equal  to 


If  there  is  a  function  U=  ^>(^,  x^  •  •  •,  xk,  >  •  •,  y^  y^  -  •  •, 
yk,  •  -  -,    2r  z2,  •  •  .,    zt,  •  -  •)  such  that 

cU_  cU_  cU 

—  •**•*>  ^     —  -*  *'         ~     —  ™» 

cxk  cijk  czk 

then  2  [Xt  dxt  +  1^  dyk  +  Zk  dz^\  is  an  exact  differential  and 
U=  ^(Xf  x^,  -  •  •,  yf  y.2,  •  -  -,  2r  2a,  •  •  •)  is  its  indefinite  integral 
and  is  the  force  function  ;  the  forces  are  a  conservative  set  ;  and 


CHAP.  IV]  POTENTIAL  ENERGY  83 

the  work  done  by  the  forces  as  the  system  moves  from  a  given 
initial  to  a  given  final  configuration  is  equal  to  the  value  of 
the  force  function  in  the  final  configuration  minus  its  value  in 
the  initial  configuration,  no  matter  by  what  paths  the  particles 
may  have  moved  from  their  initial  to  their  final  positions. 

It  is  well  known  and  can  be  shown  without  difficulty  that 
such  forces  as  gravity,  the  attraction  of  gravitation,  any  mutual 
attraction  or  repulsion  between  particles  of  a  system  which  for 
every  pair  of  particles  acts  in  the  line  joining  the  particles  and 
is  a  function  of  their  distance  apart,  are  conservative ;  while  such 
forces  as  friction,  or  the  resistance  of  the  air  or  of  a  liquid  to 
the  motion  of  a  set  of  particles,  are  not  conservative. 

•  The  negative  of  the  force  function  of  a  system  moving  under 
conservative  forces  is  called  the  potential  energy  of  the  system, 
and  we  shall  represent  it  by  V. 

If  we  are  dealing  with  motion  under  conservative  forces  and 
are  using  generalized  coordinates,  and  the  geometrical  equations 
do  not  contain  the  time,  we  can  replace  the  rectangular  coordi- 
nates of  the  separate  particles  of  the  system  in  the  force  function 
or  in  the  potential  energy  by  their  values  in  terms  of  the  gener- 
alized coordinates  q^  q2,  •  •  •,  qn,  and  we  can  thus  get  U,  and  con- 
sequently F",  expressed  in  terms  of  the  generalized  coordinates. 

If  U  is  thus  expressed,  Q^qk  (the  work  done  by  the  impressed 
forces  when  the  system  is  displaced  by  changing  qt  by  8qk^)  is  8,ftU 

and  therefore  is  approximately  —  SoA.,  or 8qk,  and  hence 

a&  cqt 

Q  =?*L=  -  — 
s&~      %*' 

39.  The  Lagrangian  and  the  Hamiltonian  Functions.  If  the 
forces  are  conservative,  our  Lagrangian  equation 

dcT±    a^_ 

dttqk       cqk~^ 

may  be  written  *-°*s.=  ™±-*L,  (2) 

dt  cqk        cqk       cqk 


84  CONSERVATIVE  FORCES  [ART.  39 

where  V  is  the  potential  energy  expressed  in  terms  of  the  coordi- 
nates q^  q2,  •  •  •,  and  not  containing  the  velocities  ^  ,  q.2,  •  •  •. 

If  L=T.-V,  (3) 

L  is  an  explicit  function  of  the  coordinates  and  the  velocities 
and  is  called  the  Lagrangian  function. 

dL      dT^          ,     dL      BTj      dV 
Obviously          —-  =  _-%    and      _  =  _?_-  — 

%      C(lk  0fc      3fc      3fc 

Hence  our  Lagrangian  equation  (1)  can  be  written  very  neatly 

££  , 

' 


If  our  forces  are  conservative,  and  we  are  using  the  Hamil- 
tonian  equations  and  express  the  kinetic  energy  Tp  in  terms  of 
the  coordinates  and  the  corresponding  momenta,  and  if  we  let 

ff=Tf  +  r,  (5) 

If  is  called  the  Hamiltonian  function  and  is  a  function  of  q^  q^ 
...,  qn,  and^,^a,  .  -  -,  pn. 
Our  Hamiltonian  equations 


can  now  be  written 

dp,  =  _dH 

dt  cqk 

dqk_8H 

d£-~fyh' 

and  these  are  known  as  the  Hamiltonian  canonical  equations. 

If  our  forces  are  conservative,  and  we  are  using  instead  of 
the  kinetic  energy  the  Lagrangian  expression  for  the  energy 
modified  for  some  of  the  coordinates  q^  <?2,  •  •  •,  qr  (v.  Art.  17), 
and  if  we  let  ^  ^  (g) 


CHAP.  IV]          THE  LAGRANGIAN  FUNCTION  85 

we  have  for  any  coordinate  qr+t  the  Lagrangian  equation 


and  for  any  coordinate  qr_k  the  pair  of  Hamiltonian  equations 
dpr_k  _   d<£  n  n, 

^r~%r; 

dqr-t_        d® 

-      - 


40.  The  Lagrangian  function  Z.  is  the  difference  between  the 
kinetic  energy  T^  (expressed  in  terms  of  the  coordinates  q^  q2, 
'  •  •,  qn,  and  the  velocities  q^  q^  •  •  •,  qn,  and  homogeneous  of  the 
second  degree  in  terms  of  the  velocities)  and  the  potential 
energy  V  (expressed  in  terms  of  the  coordinates  q^  q2,  •  •  •,  <?„). 

The  Hamiltonian  function  H  is  the  sum  of  the  kinetic 
energy  Tp  (expressed  in  terms  of  the  coordinates  q^  q^  •  •  .,  qn, 
and  the  corresponding  momenta  p^  pz,  •  •  .,  pn,  and  homogen- 
eous of  the  second  degree  in  terms  of  the  momenta)  and 
the  potential  energy  V  (expressed  in  terms  of  the  coordinates 

&  »  <1#  '  '  •'   &)• 

The   sum   of  the  kinetic  energy  and   the   potential  energy, 

however  expressed,  is  sometimes  called  the  total  energy  of  the 
system,  and  we  shall  represent  it  by  E,  so  that 

E=T+V.  (1) 

The  function  <I>  of  the  preceding  section  is  the  difference 
between  the  kinetic  energy  (expressed  in  terms  of  the  coordi- 
nates q^  q2,  -  •  -  ,  qn,  the  momenta  p^  p2,  •  •  •  ,  j»r,  and  the 
velocities  qr+l,  qr+2,  •  •  •,  ^B),  minus  the  terms 


(similarly  expressed),  and  the  potential  energy  (expressed  in 
terms  of  the  coordinates  q^  q2,  •  •  •  ,  qn*).  We  shall  call  it  the 
modified  Lagrangian  function. 


86  CONSERVATIVE  FORCES  [ART.  41 

It  is  to  be  observed  that  all  the  terms  of  <I>  except  those 
contributed  by  the  potential  energy  are  homogeneous  of  the 
second  degree  in  the  momenta  introduced  and  the  velocities  not 
eliminated  by  the  modification. 

41.  In  dealing  with  the  motion  of  a  system  under  conserva- 
tive forces,  we  may  form  the  differential  equations  of  motion 
in  any  one  of  three  ways,  and  the  equations  in  question  are 
practically  given  by  giving  a  single  function  —  L,  the  Lagrangian 
function,  or  H,  the  Hamiltonian  function,  or  4>,  the  modified 
Lagrangian  function.* 

Every  one  of  these  functions  consists  of  two  very  different 
parts :  one,  the  potential  energy  F,  which  depends  merely  on  the 
forces,  which  in  turn  depend  solely  on  the  configuration  of  the 
system;  the  other,  the  kinetic  energy  T  or  the  modified 
Lagrangian  expression  Mq  ...,  either  of  which  involves  the  veloci- 
ties or  the  momenta  of  the  system  as  well  as  its  configuration. 

If  we  are  using  as  many  independent  coordinates  as  there  are 
degrees  of-  freedom,  a  mere  inspection  of  the  given  function 
will  enable  us  to  distinguish  between  the  two  functions  of 
which  it  is  formed,  the  potential  energy  or  its  negative  being 
composed  of  all  the  terms  not  involving  velocities  or  momenta. 

If,  however,  we  are  ignoring  some  of  the  coordinates  (v.  Arts. 
16,  21,  and  24)  and  are  using  JET  (the  Hamiltonian  function)  or  <E> 
(.the  Lagrangian  function  modified  for  the  ignored  coordinates), 
the  portion  contributed  to  H  by  Tp  or  to  <l>  by  M  (the  modified 
expression  for  the  kinetic  energy)  is  no  longer  necessarily  a 
homogeneous  quadratic  in  the  velocities  and  momenta  (v. 
Art.  21)  and  may  contain  terms  involving,  merely  the  coordi- 
nates and  therefore  indistinguishable  from  terms  belonging  to 
the  potential  energy ;  and  consequently  the  part  of  the  motion 
not  ignored  would  be  identical  with  that  which  would  be  pro- 
duced by  a  set  of  forces  quite  different  from  the  actual  forces. 

*  Indeed,  for  equations  of  the  Lagrangian  type,  any  constant  multiple  of  L 
or  <I>  will  serve  as  well  as  L  or  <f>. 


CHAP.  IV]  ILLUSTRATIVE  EXAMPLE  87 

We  may  note  that  the  last  paragraph  does  not  apply  if  the 
system  starts  from  rest,  so  that  the  ignored  momenta  are  zero 
throughout  the  motion  (v.  Art.  24,  (a)). 

Let  us  consider  the  problem  of  Art.  8,  (c?),  where  the  potential 
energy  V  is  easily  seen  to  be  —  mgx. 

If  we  are  using  the  Lagrangian  method,  as  in  Art.  8,  (cZ), 

we  have 

Z  =  |[2z2  +  («-:r)0-2  +  2^].  (1) 

If  we  use  the  Hamiltonian  method,  as  in  Art.  15,  (<i),  we 

have  i    r     2  2  -i 

P*   ,«-20tfrr  .  (2) 

(a  —  x)' 


If    we    use    the   Lagrangian    method    modified   for  6,   as  in 
Art.  20,  (6),  we  have 


x2  -  —&  -  ,  +  2  gx\. 
m\a-xf  J 


(3) 


A  mere  inspection  of  any  one  of  these  three  functions  enables 
us  to  pick  out  the  potential  energy  as  F  =  —  mgx. 

If,  however,  we  are  ignoring  #,  as  in  Art.  22,  Example,  we 

have 

(4) 


and  here,  so  far  as  the  function  <&  shows,  the  potential  energy 

c2 
may  be  —  myx,  as,  in  fact,  it  is,  or  it  may  be 


9  ™  ( n  TV2 

«  ill  (  w-  ~^  JL  j 

Indeed,  the  hanging  particle  moves  precisely  as  if  its  mass  were 
2  m  and  it  were  acted  on  by  a  force  having  the  force  function 

U=-  ~--0  +  mgx; 

2m(a  —  xy 

(? 
that  is,  a  force  vertically  downward  equal  to  mg -g. 

7/fr  (  CL  ™ "™*  3x  ) 

If,  as  in  many  important  problems  (v.  Chap.  V),  we  are  unable 
to  discern  and  measure  the  impressed  forces  directly  and  are 
attempting  to  deduce  them  from  observations  on  the  behavior  of 


88  CONSERVATIVE  FORCES  [Airr.  42 

a  complicated  system,  which  for  aught  we  know  may  contain 
undetected  moving  masses,  the  fact  that  we  cannot  discriminate 
with  certainty  between  the  terms  contributed  to  the  modified 
Lagrangian  function  by  the  kinetic  energy  of  the  system  and 
those  contributed  by  the  potential  energy  may  lead  to  entirely 
different  and  equally  plausible  explanations  of  the  observed 
phenomena  (v.  Art.  51). 

42.  Conservation  of  Energy.  If  we  are  dealing  with  a  system 
moving  under  conservative  forces,  the  coordinates  q^  q^  •  •  •  ,  qn, 
are  functions  of  t,  the  time,  as  are  also  the  velocities  q^  <?2,  •  •  •  ,  qn. 
Therefore  F",  the  potential  energy,  and  T,  the  kinetic  energy,  are 
functions  of  the  time,  as  is  their  sum,  the  Hamiltonian  function  H. 

«     A    dH 

Let  us  find  -—  • 
dt 

As  .#"  depends  explicitly  on  the  coordinates  qlt  q^  •  •  •,  qn,  and 
the  momenta  p^  pz,  •  •  •,  pn, 

—  ^ 


But  by  our  Hamiltonian  equations  (v.  Art.  39,  (6)  and  (7)), 

dp,.         cH  ,     do,.      cH 

-~  =  -  -—  »     and     -fi  =  -—  , 
at  cqk  at       opk 

and  (1)  becomes 


qk    pk        pk  cqk 

Therefore  T  +  V  =  H  =  h,  (3) 

where  h  is  a  constant. 

Hence  in  any  system  moving  under  conservative  forces  the 
sum  of  the  kinetic  energy  and  the  potential  energy  is  constant 
during  the  motion. 

This  is  called  the  Principle  of  the  Conservation  of  Energy. 

Since  by  (3)  any  loss  in  potential  energy  during  the  motion 
is  just  balanced  by  an  increase  in  the  kinetic  energy,  and  the 
loss  in  potential  energy  is  equal  to  the  work  done  by  the  actual 


CHAP.  IV]  HAMILTON'S  PRINCIPLE  89 

forces  during  the  motion,  our  principle  is  a  narrower  statement 
of  the  familiar  principle  :  If  a  system  is  moving  under  any  forces, 
conservative  or  not,  the  gain  in  kinetic  energy  is  always  equal  to 
the  work  done  by  the  actual  forces. 

43.  Hamilton's  Principle.  Let  a  system  move  under  conserv- 
ative forces  from  its  configuration  at  the  time  tQ  to  its  con- 
figuration at  the  time  t^  We  have 

d  cL  _  dL  _ 

dtcqk      cqk' 

where  L,  the  Lagrangian  function,  is  equal  to  T  —  V. 

Suppose  that  the  system  had  been  made  to  move  from  the 
first  to  the  second  configuration  so  that  the  particles  traced 
slightly  different  paths  with  slightly  different  velocities,  but  so 
that  at  any  time  t  every  coordinate  qk  differed  from  its  value 
in  the  actual  motion  by  an  infinitesimal  amount,  and  so  that 
every  velocity  qk  differed  from  its  value  in  the  actual  motion 
by  an  infinitesimal  amount,  or  (using  the  notation  of  the 
calculus  of  variations}*  so  that  8qk  and  8qk  were  infinitesimal  ; 
and  suppose  it  had  reached  the  second  configuration  at  the 
time  ^.  Then,  if  at  the  time  t  the  difference  between  the  value 
of  L  in  the  hypothetical  motion  and  its  value  in  the  actual 
motion  is  Si, 


,  at  the  time  t, 


dL  -,  .        dL  d 

and  —       =  -    - 


=  d_/dL_      \_d__ 


d  IcL  .    \      cL  £       , 

"^*9"  y 


*  For  a  brief  introduction  to  the  calculus  of  variations,  see  Appendix  B. 


90  CONSERVATIVE  FORCES  [ART.  44 


Therefore,  by  (2),  SL  =    -V 

dt  **  [cqk 

and  C\Ldt  =  8  f  LM  =  [  Y  —  So,]*1. 

A  A  L^*      J'. 


Since  the  terminal  configurations  are  the  same  in  the  actual 
motion  and  in  the  hypothetical  motion  and  the  time  of  transit 
is  the  same,  8qk  =  0  when  t  =  tQ  and  when  t  =  t^  and 

S  flLdt  =  8  f  '[T-F]cft=0.  (3) 

A  A 

But  (3)  is  the  necessary  condition  that  L  should  be  either  a 
maximum  or  a  minimum  and  is  sometimes  stated  as  follows: 
When  a  system  is  moving  under  conservative  forces,  the  time  inte- 
gral of  the  difference  between  the  kinetic  energy  and  the  potential 
energy  of  the  system  is  "  stationary  "  in  the  actual  motion.  This 
is  known  as  Hamilton's  principle.* 

44.  The  Principle  of  Least  Action.  If  the  limitation  in  the  pre- 
ceding section  that  "  in  the  actual  and  the  hypothetical  motions 
the  time  of  transit  from  the  first  to  the  second  configuration  is 

*  Hamilton's  principle  plays  so  important  a  part  in  mechanics  and  physics 
that  it  seems  worth  while  to  obtain  a  formula  for  it  which  is  not  restricted  to 
conservative  systems. 

We  shall  use  rectangular  coordinates,  and  we  shall  make  the  hypothesis  as  to 
the  actual  motion  and  the  hypothetical  motion  which  has  been  employed  above. 

For  every  particle  of  the  system  we  have  the  familiar  equations 

mx  =  -5T,        my  =  F,        mz  =  Z. 
Since  T=  ^  7|(x2  +  #2  +  z2), 

ST=  Sm[x5x  +  ydy  +  fSz]  =  2m|x  —  5x  +  y  —  5y  +  z  —  5z\- 

L   dt  dt  dt     J 

But      mx  —  Sx  =;  TO  —  (xSx)  —  mxSx  —  —  (mxdx)  —  A"5x. 
dt  dt  dt 

Hence   5T  +  S  [XSx  +  Ydy  +  ZSz]  =  -  2m  [x5x  +  ySy  +  zSz], 

and          f  '1{ST  +  S  [JTSx  +  Ydy  +  ZSz]  }  dt  =  S  [mxSx  +  ySy  +  z5z#  =  0. 

J'o  ° 

If  the  system  is  conservative,  S  [JTSx  +  Ydy  +  ZSz]  =  SU  =  —  SF,  and  we 
get  the  formula  (3)  in  the  text. 


CHAP.  IV]          PRINCIPLE  OF  LEAST  ACTION  91 

the  same  "  be  removed  and  the  variations  be  taken  not  at  the 
same  time  but  at  arbitrarily  corresponding  times,  t  will  no  longer 
be  the  independent  variable  but  will  be  regarded  as  depending 
upon  some  independent  parameter  r.  We  now  have  (v.  Art.  43) 

%  =  ^  %  -  &  ^  ^        (v.  APP-  B»  §  6»  C1)) 
dL  *  .       $L  d    ,          .  dL  d  ~  , 


d  /dL  ,    \      dL  ,         .  dL  d 

=  —  (  —  oq,.  )  --  og,.  —  q^  -- 

dt\dqk    **}      cqk    ^         *kcqkdt 


and 


since 

Hence 
and 


f 


* 


If  now  we  impose  the  condition  that  during  the  hypothetical 
motion,  as  during  the  actual  motion,  the  equation  of  the  con- 
servation of  energy  holds  good,  that  is,  that 


then 

S 
and  (1)  becomes 


92  CONSERVATIVE  FOKCES  [ART.  45 


8  (2  T)  +  2  TS  —1  rfr  =  0, 
<frj 


fW     i  ftor,  *' 


f** 

and,  finally,  8  T  22Vfe  =  0.  (  2  i 

«/(„ 

/•"i  &dk> 

The  equation  -^  =  /     2  Tcfa  defines  the  action,  A,  of  the  forces, 


and  the  fact  stated  in  (2)  is  usually  called  the  principle  of  least 
action.  As  a  matter  of  fact,  (2)  shows  merely  that  the  action  is 
"  stationary." 

45.  In  establishing  Hamilton's  principle  we  have  supposed 
the  course  followed  by  the  system  to  be  varied,  subject  merely 
to  the  limitation  that  the  time  of  transit  from  initial  to  final 
configuration  should  be  unaltered  ;  consequently,  as  the  total 
energy  is  not  conserved,  the  varied  course  is  not  a  natural 
course.  That  is,  to  compel  the  system  to  follow  such  a  course 
we  should  have  to  introduce  additional  forces  that  would 
do  work. 

In  establishing  the  principle  of  least  action,  however,  we 
have  supposed  the  course  followed  by  the  system  to  be  varied, 
subject  to  the  limitation  that  the  total  energy  should  be  unal- 
tered ;  consequently  the  varied  course  is  a  natural  course.  That 
is,  to  compel  the  system  to  follow  such  a  course  we  need 
introduce  merely  suitable  constraints  that  would  do  no  work. 

We  have  deduced  both  principles  from  the  equations  of 
motion.  Conversely,  the  equations  of  motion  can  be  deduced 
from  either  of  them.  Each  of  them  is  therefore  a  necessary 
and  sufficient  condition  for  the  equations  of  motion. 


CHAP.  IV]  DEFINITIONS  OF  ACTION  93 

46.  The  action,  A,  of  a  conservative  set  of  forces  acting  on  a 
moving  system  has  been  defined  as  the  time  integral  of  twice 
the  kinetic  energy. 

A  =  \    l2Tdt.  (1) 

J  tt, 

r'i  r'i       /fj<t\s         r*i 

But     A=  I    2m(x2  +  f  +  z2)dt  =        Zmi  —  ]dt=  I    2mv2dt. 
Jt0  Jt0  V**/  Jt0 

Therefore  A  =  /  ^.rnvcls,  (2) 

so  that  the  action  might  just  as  well  have  been  defined  as  the 
sum  of  terms  any  one  of  which  is  the  line  integral  of  the  momen- 
tum of  a  particle  taken  along  the  actual  path  of  the  particle. 

There  is  another  interesting  expression  for  the  action,  which 
does  not  involve  the  time  even  through  a  velocity. 

Since  T+V=h,          2T=2(A-F). 


=  2  (A  -  F)  dt\ 

* 


A  =      vC/i-  V}^mds\  (3) 

47.  We  have  stated  without  proof  that  the  differential  equa- 
tions of  motion  for  any  system  under  conservative  forces  could 
be  deduced  from  Hamilton's  principle  or  from  the  principle  of 
stationary  action.  Instead  of  giving  the  proof  in  general,  we  will 
give  it  in  a  concrete  case,  that  of  a  projectile  under  gravity. 


94  CONSERVATIVE  FORCES  [ART.  47 

We  shall  use  fixed  rectangular  coordinates,  T  horizontal  and 
X  vertically  downward,  taking  the  origin  at  the  starting  point 
of  the  projectile.  In  this  case,  obviously, 


or 


),     and      V  =  -mgx. 
(a)  By  Hamilton's  principle,  we  have 

/**i  rm 

8  I     |[^  +  /  +  2^]^  =  0,  (1) 

Jo       ^ 

C18[x*  +  f  +  2gx']dt  =  Q. 

Jo 

r'i 

I     \jcbx  +  y§y  +  g$x]  dt  =  0. 

Jo 


Integrating  by  parts, 


but  since  Sa:  and  8y  are  wholly  arbitrary,  this  is  impossible  unless 
the  coefficients  of  8x  and  §y  in  the  integrand  are  both  zero. 

d?x 
Hence  —  -^r  =  0, 

' 


By  the  pruiciple  of  stationary  action,  we  have 


whence 


/*ri  r//  /^ri  ^7/ 

I     (x2  +  y2)3^^r  +  /      ^-S(x*  +  f-)dr  =  0.        (1) 
Jr.  dr         Jra    dr 


CHAP.  IV]  LEAST  ACTION  95 

But  —  (1?  +  #2)  —  mgx  =  L 


Hence 
and 


g  ^ff  =        j££       +       jty==j._8X       +       g_8y_        ^       +      ^        __       ^ 

since  ^=s      =      &r-fc;     (v.  App.  B,  §  6,  (1)) 

8t', 
dr 


C r*  r    d  d      ~\  C r*        dt 

or  I      \x  —  Sx  +  y  — -  Sy  \dr  +  I      gSx  —  dr  =  Q. 

Jro    I    dr  dr      \         J,.o    '       dr 

Integrating  by  parts,  we  get 

[~|ri       CTl\  ••  dt  ..dt  dt     ~\ 

x8x  -j-  y$y  I    —  I     I  x  —  8x  -\-  y  —  $y  —  g  —  8x  dr  =  0. 
Jra     Jro    I   dr  dr  dr-    \ 

But  Sx  and  8y  are  zero  at  the  beginning  and  at  the  end  of 
the  actual  and  the  hypothetical  paths,  so  that 

/"•i     ..  >g       fa 

Jr.  '   dr 

and  as  Bx  and  8y  are  wholly  arbitrary,  this  necessitates  that 


as  in  Art.  47,  (a). 


96  CONSEEVATIVE  FORCES  [ART.  48 

48.  Although  we  have  proved  merely  that  in  a  system  under 
conservative  forces  the  action  satisfies  the  necessary  condition 
of  a  minimum,  namely,  8 A  =  0,  it  may  be  shown  by  an  elabo- 
rate investigation  that  in  most  cases  it  actually  is  a  minimum, 
and  that  the  name  "  least  action,"  usually  associated  with  the 
principle,  is  justified. 

A  very  pretty  corollary  of  the  principle  comes  from  its  applica- 
tion to  a  system  moving  under  no  forces  or  under  constraining1 
forces  that  do  no  work.    In  either  of  these  cases  the  potential 
energy  V  is  zero,  and  consequently  the  kinetic  energy  is  constant, 
that  is,  T=  h. 

A  =  C  12 Tdt  =  2  h  C  'dt  =  2  h^  -  Q 

•A,  -A 

and  the  action  is  proportional  to  the  time  of  transit.  Hence 
the  actual  motion  is  along  the  course  which  occupies  the  least 
possible  time. 

For  instance,   if  a  particle  is  moving  under  no  forces,  the 

/Yft 

energy  —  v2  is  constant,  the  velocity  of  the  particle  is  constant, 
& 

and  as  it  moves  from  start  to  finish  in  the  least  possible  time, 
it  must  move  from  start  to  finish  by  the  shortest  possible  path, 
that  is,  by  a  straight  line. 

If  instead  of  moving  freely  the  particle  is  constrained  to 
move  on  a  given  surface,  the  same  argument  proves  that  it 
must  trace  a  geodetic  on  the  surface. 

49.  Varying  Action.*    The  action,  A,  between  two  configura- 
tions  of   a   system  under   conservative   forces   is   theoretically 
expressible  in  terms  of  the  initial  and  final  coordinates  and  the 
total  energy  (v.  Art.  46,  (3)),  and,  when  so  expressed,  Hamilton 
called  it  the  characteristic  function. 

In  like  manner,  if 

r  (i  r'i  r'i 

S=  I    Ldt=  I     (T-V)dt  =        (li-2 

J'0        A  »A> 

*  For  a  detailed  account  of  Hamilton's  method,  see  Routh,  Advanced  Rigid 
Dynamics,  chap,  x,  or  Webster,  Dynamics,  §  41. 


CHAP.  IV]  VARYING  ACTION  97 

S  may  be  expressed  in  terms  of  the  initial  and  final  coordinates 
of  the  system,  the  total  energy,  and  the  time  of  transit.  When 
so  expressed,  Hamilton  called  it  the  principal  function.  By 
considering  the  variation  produced  in  either  of  these  functions 
by  varying  the  final  configuration  of  the  system,  Hamilton 
showed  that  from  either  of  these  functions  the  integrals  of 
the  differential  equations  of  motion  for  the  moving  system 
could  be  obtained,  and  he  discovered  a  partial  differential 
equation  of  the  first  order  that  S  would  satisfy  and  one  that  A 
would  satisfy,  and  so  reduced  the  problem  of  solving  the  equa- 
tions of  motion  for  any  conservative  system  to  the  solution  of 
a  partial  differential  equation  of  the  first  order. 

It  must  be  confessed,  however,  that  in  most  cases  the  advan- 
tage thus  gained  is  theoretical  rather  than  practical,  as  the 
solving  of  the  equation  for  S  or  for  A  is  apt  to  be  at  least  as 
difficult  as  the  direct  solving  of  the  equations  of  motion. 


CHAPTER  V 
APPLICATION  TO  PHYSICS 

50.  Concealed  Bodies.     In  many  problems  in   mechanics  the 
configuration  of  the  system  is  completely  known,  so  that  a  set 
of  coordinates  can  be  chosen  that  will  fix  the  configuration 
completely  at  any  time;  and  the  forces  are  given,  so  that  if 
the  system  is  conservative  the  potential  energy  can  be  found. 
In  that  case  the   Lagrangian  function  L  or  the  Hamiltonian 
function  H  can  be  formed,  and  then  the  problem  of  the  motion 
of  the  system  can  be  solved  completely  by  forming  and  solving 
the  equations  of  motion. 

In  most  problems  in  physics,  however,  and  in  some  problems 
in  mechanics  the  state  of  things  is  altogether  different.  It  may 
be  impossible  to  know  the  configuration  or  the  forces  in  their 
entirety,  so  that  the  choice  of  a  complete  set  of  coordinates  or 
the  accurate  forming  of  the  potential  function  is  beyond  OUT 
powers,  while  it  is  possible  to  observe,  to  measure,  and  partly  to 
control  the  phenomena  exhibited  by  the  moving  system  which 
we  are  studying.  If  from  results  which  have  been  observed  or 
have  been  deduced  from  experiment  we  are  able  to  set  up  in- 
directly the  Lagrangian  function,  or  the  Hamiltonian  function, 
or  the  modified  Lagrangian  function,  we  can  then  form  our  dif- 
ferential equations  and  use  them  with  confidence  and  profit. 

51.  Take,  for  instance,  the  motion  considered  in  the  problem 
of  the  two  equal  particles  and  the  table  with  a  hole  in  it  (v. 
Art.  8,  (£?))>  and  suppose  the  investigator  placed  beneath  the 
table  and  provided  with  the  tools  of  his  trade,  but  unable  to 
see  what  is  going  on  above  the  surface  of  the  table.    He  sees 
the   hanging    particle   and   is   able  to  determine   its  mass,   its 


CHAP.  V]  THE  DANGLING  PAKTICLE  99 

velocity,  and  its  acceleration ;  to  fix  its  position  of  equilibrium ; 
to  measure  its  motion  under  various  conditions ;  to  apply  addi- 
tional forces,  finite  or  impulsive,  and  to  note  their  effects. 

His  system  has  apparently  one  degree  of  freedom.  It  pos- 
sesses an  apparent  mass  m  and  is  certainly  acted  on  by  gravity 
with  a  downward  force  mg.  He  determines  its  position  of 
equilibrium,  and  taking  as  his  coordinate  its  distance  x  below 
that  position,  he  painfully  and  laboriously  finds  that  x,  the  accel- 
eration of  the  hanging  mass,  is  equal  to 


ir 
4 


He  is  now  ready  to  call  into  play  his  knowledge  of  mechanics. 

If  T  is  the  kinetic  energy  and  L  is  the  Lagrangian  function  for 

the  system  which  he  sees, 


T  -  — 


and  L=T-  V=x*- 


d  (dL\      8L 

Since  _(_)  =  — 

at  \ox/      ox 

.._3L_      dV  1 

""  -       = 


rn       t  %V     m 

Therefore 


ex       2    (a  -  x)* 
mg[       a3  "1      mg[       as 

V  =  2  [W^f  -'  \  =  2  [20^? 

The    motion,   then,    can   be    accounted    for    as    due  to  the 
downward  force  of  gravity  combined  with  a  second  vertical 

force  having  the  potential  energy  -^-  -  -  —2  +  x  ,  that  is,  a 

r          s  ~~\ 

force  vertically  upward  having  the  intensity  -^-  --  ^  +  1  1  ; 

2  \_(a  —  x)         J 

and  of  course  this  force  must  be  the  pull  of  the  string  and  may 
be  due  to  the  action  of  some  concealed  set  of  springs. 


100  APPLICATION  TO  PHYSICS  [ART.  51 

On  the  other  hand,  the  moving  system  may  contain  some 
concealed  body  or  bodies  in  motion,  and  that  this  is  the  fact  is 
strongly  suggested  if  a  downward  impulsive  force  P  is  applied 
to  the  hanging  particle  when  at  rest  in  its  position  of  equilib- 
rium. For  such  a  force  is  found  to  impart  an  instantaneous 

p 
velocity  x  =  -  —  ,  just  half  what  we  should  get  if  the  hanging 

particle  were  the  only  body  in  the  system,  and  just  what  we 
should  have  if  there  were  a  second  body  of  mass  m  above  the 
table,  connected  with  the  hanging  particle  by  a  stretched  string 
of  fixed  length. 

Obviously  this  concealed  body  is  ignorable,  since  we  have 
already  found  a  function  from  which  we  can  obtain  the  differ- 
ential equation  for  x,  namely,  the  function  we  have  just  called 
the  Lagrangian  function  L.  It  is 


and  contains  the  single  coordinate  x. 

But  if  we  have  ignored  a  concealed  moving  body  form- 
ing part  of  our  system,  this  expression  is  not  the  Lagrangian 
function  L,  but  is  equal  or  proportional  to  <l>,  the  Lagrangian 
function  modified  for  the  coordinate  or  coordinates  of  the  con- 
cealed body  ;  and  in  that  case  some  or  all  of  the  terms  which 
we  have  regarded  as  representing  the  potential  energy  of  the 
system  may  be  due  to  the  kinetic  energy  of  the  concealed  body 
(v.  Art.  41). 

Of  course  the  complete  system  may  have  two  or  more  degrees 
of  freedom.  Let  us  see  what  we  can  do  with  two.  Take  x 
and  a  second  coordinate  6  and  remember  that  as  6  is  ignor- 
able it  must  be  a  cyclic  coordinate  and  must  not  enter  into 
the  potential  energy. 

Let  us  now  form  the  Lagrangian  function  and  modify  it  for  6. 

We  have  T  =  Ax2  -f-  Bxd  +  C02,  where  A,  B,  and  C  are  functions 
of  x. 


CHAP.  V]  THE  DANGLING  PARTICLE  101 


To  modify  for  6  we  must  subtract  8pe.    We  get 

B 

Me  =  Ax  +  — 

i-'-i 

Suppose  that  no  external  force  acts  on  the  concealed  particle, 
so  that  the  potential  energy  of  the  system  is  —  mgx. 
Then,  if  the  Lagrangian  function  modified  for  6  is  <l>, 

[B2 1          B  p2 

A~Tc\^^~  '%cp*  ~  4c+mgx' 

Since    on    our  ignoration   hypothesis   6   does   not   enter   the 
potential  energy,  the  momentum  p6  =  K,  a  constant,  and 


But  we  know  that  4>  is  equal  or  proportional  to 
m  .2      mg        az  mgx 

the  function  which  on  our  hypothesis  of  no  concealed  bodies 
we  called  L. 

We  see  that  if  B  =  0,  if  A  =  m,  and  if  - —  =  - — , 

T!    O  —   ^   '/    ZC  J 

whence  C  —  — 


2       mga8 

Then  we  have  T  =  mx2  •+- ^ 

2       mga8 


102  APPLICATION  TO  PHYSICS  [ART.  51 

where  K  is  the  momentum  corresponding  to  the  coordinate  6 
and  may  be  any  constant. 
If  we  take  for  K2  the  value 


is  obviously  the  kinetic  energy  of  a  mass  m  moving  in  a  plane 
and  having  a  —  x  and  0  as  polar  coordinates.  The  Lagrangian 
function  is  equal  to 


and  the  x  Lagrangian  equation  is 

2  mx  =  —  m(a  —  x)&2  +  mg.  (1) 

If  the  angular  velocity  of  the  concealed  mass  when  the  hang- 
ing particle  is  at  rest  in  its  position  of  equilibrium  is  #0,  since 

in  that  case  x  =  0  and  x  =  0,  equation  (1)  gives  us  #0  =  -vl-  •    The 

observed  motion  of  the  hanging  particle  is  then  accounted  for 
completely  by  the  hypothesis  that  it  is  attached  by  a  string 
to  an  equal  particle  revolving  on  the  table  and  describing  a 
circle  of  radius  a  about  the  hole  in  the  table  with  angular 

velocity  -\l-  when  the  hanging  particle  is  at  rest  in  its  posi- 

i  Ct 

tion  of  equilibrium.    We  see  that  on  this  hypothesis  the  term 

o     Q  -  2  +  x  '  wm°h  on  the  hypothesis  that  the  system 
A   I  _  (  tt  —  Xj 

contained  only  the  hanging  particle  was  an  unforeseen  part  of 
the  potential  energy,  is  due  to  the  kinetic  energy  of  the  con- 
cealed moving  body. 

It  may  seem  that  giving  K  a  different  value  might  lead  to- 
a  different  hypothesis  as.  to  the  motion  of  the  concealed  body 
that  would  account  for  the  motion  of  the  hanging  particle. 


CHAP.  V]  PHYSICAL  COORDINATES  103 

Such,  however,  is  not  the  case.    We  have 


2 

Let  </>  =  —  ==  6. 

mvgcf 

it/  • 

Then  T  =  ^  [z2  -j-  z2  +  (a  -  z)2  <j>2]. 

M 

Ttl/  * 

—  \y?  -j-  (a  —  #)2  <£2]  is,  as  above,  the  kinetic  energy  of  a  body 

A 

of  mass  m,  with  polar  coordinates  a  —  x  and  (f>  ;  and  the  concealed 
motion  is  precisely  as  before.  In  using  the  form  (2)  we  have 
merely  used  as  our  second  generalized  coordinate  #,  a  perfectly 
suitable  parameter  but  one  less  simple  than  the  polar  angle. 

52.  Problems  in  Physics.    In  physical  problems  there  may  be 
present    electrical    and    magnetic    phenomena    and    concealed 
molecular  motions,  as  well  as  the  visible  motions  of  the  mate- 
rial parts  of  the  system.    In  such  cases,  to  fix  the  configuration 
of  the  system  even  so  far  as  it  is  capable  of  being  directly 
observed   we   must   employ  not   only  geometrical   coordinates 
required  to  fix  the  positions  of  its  material  constituents,  but  also 
parameters  that  will  fix  its  electrical  or  magnetic  state  ;  and  as 
we    are    rarely   sure   of    the   absence    of   concealed   molecular 
motions,  we  must  often  allow  for  the  probability  that  the  func- 
tion we   are  trying  to   form   by   the   aid   of   observation    and 
experiment  and  on  which  we  are  to  base  our  Lagrangian  equa- 
tions  of    motion   may   be   the    Lagrangian    function    modified 
for   the    ignored   coordinates   corresponding    to    the    concealed 
molecular  motions. 

53.  Suppose,  for  instance,  that  we  have  two  similar,  parallel, 
straight,  conducting  wires,  through  which  electric  currents  due 
to  applied  electromotive  forces  are  flowing.     It  is  found  ex- 
perimentally that  the  wires  attract  each  other  if  the  currents 
have  the   same   direction,  and  repel  each   other  if  they  have 
opposite   directions,   and  that  reversing   the   currents   without 


104  APPLICATION  TO  PHYSICS  [ART.  53 

altering  the  strength  of  the  applied  electromotive  forces  does 
not  affect  the  observed  attraction  or  repulsion.  It  is  known 
that  an  electromotive  force  drives  a  current  against  the  resist- 
ance of  the  conductor,  and  that  the  intensity  of  the  current  is 
proportional  to  the  electromotive  force.  Moreover,  it  is  known 
that  an  electromotive  force  does  not  directly  cause  any  motion 
of  the  conductor.  It  is  found  that,  as  far  as  electric  currents 
are  concerned,  the  phenomena  depend  merely  on  the  intensity 
and  direction  of  the  currents. 

To  fix  our  configuration  we  shall  take  x  as  the  distance 
between  the  wires  and  take  parameters  to  fix  the  intensities  of 
the  two  currents.  These  parameters  might  be  regarded  as 
coordinates  or  as  generalized  velocities,  but  many  experiments 
suggest  that  they  are  velocities.  We  shall  call  them  yv  and  f/n 
and  define  yt  as  the  number  of  units  of  electricity  that  have 
crossed  a  right  section  of  the  first  wire  since  a  given  epoch. 

As  all  effects  depend  upon  y^  and  yz  and  not  on  yl  and  «/2, 
yl  and  yz  are  cyclic  coordinates. 

Let  us  suppose  that  there  are  no  concealed  motions.  Then 
the  kinetic  energy  T  is  a  homogeneous  quadratic  in  x,  y^  and 

y .    Let 

T  =  Ax*  +  Lyl  +  Myj,  +  Ntf  +  Bx^  +  Cxyf          (1) 

where  the  coefficients  are  functions  of  x. 

Since  reversing  the  directions  of  the  currents,  that  is,  revers- 
ing the  signs  of  y^  and  y2,  does  not  change  the  other  phenomena, 
it  must  not  affect  T;  therefore  B  and  C  are  zero. 

If  the  wires  are  not  allowed  to  move,  x  =  0  and  T  reduces 
to  Lyl  +  My^yz  +  Ny%,  which  is  called  the  electrokinetic  energy 
of  the  system.  Since  from  considerations  of  symmetry  this  can- 
not be  altered  by  interchanging  the  currents,  N—L. 

Let  us  now  suppose  that  the  first  wire  is  fastened  in  position 
and  that  the  only  external  forces  are  the  electromotive  forces 
El  and  Ez,  producing  the  currents  in  the  two  wires ;  the  resist- 
ances Ry^  and  Ri/a  of  the  wires,  equal,  respectively,  to  E^  and 
jE"2  when  the  currents  yv  and  yz  are  steady ;  and  an  ordinary 


CHAP.  V]        PAKALLEL    LINEAR  CONDUCTORS  105 

mechanical  force  F,  tending  to  separate  the  wires.    We  now 

have  o  ™ 

v™      0  ,  . 
P*  =  -^  =  2Ax, 

dx 

and  for  our  x  Lagrangian  equation 

ndA  .o      dA  .„      dL  .        dM  .  dL  .„ 

2^  +  2_*'--i._-y.-_^l-_j.-J.    (2) 

Let  us  study  this  equation.  First  suppose  the  electromotive 
forces  and  the  impressed  force  F  all  zero,  so  that  y^  =  yz  =  0, 
and  F=Q.  Equation  (2)  reduces  to 

—  xz  =  0, 
dx 

1    dA  .. 

or  x  =  -  —  —  x2. 

2  A  dx 

If,  then,  a  transverse  velocity  were  impressed  on  the  second 

dA 

wire,  the  wire  would  have  an  acceleration  unless  -—  =  0.    But 

dx 

both  wires,  on  our  hypothesis,  being  inert,  they  can  neither 
attract  nor  repel  each  other,  therefore  A  is  a  constant  ;  and  as 
T  =  Ax*  when  y^  =  yz  =  0,  A  is  a  positive  constant.  Therefore 
(2)  reduces  to  • 


Let  us  now  suppose  that  yz  =  0,  and  F=Q.    Equation  (3) 

becomes  2  Ax  —  —  y\  =  0, 

dx 

1    dL  ., 

or  x  =  -——y\, 

2  A  dx 

and    the    second   wire    is    attracted    or   repelled  by  the  first, 
even  when   no   current   is   flowing   through  the  second   wire. 

This  is  contrary  to   observation.     Therefore   —  =  0,  and  L  is 

(A/J(/ 

a  constant;   and  as  T=Lyl  when  x  and  y^  are  zero,  L  is  a 


106  APPLICATION  TO  PHYSICS  [ART.  51 

positive  constant.    Equation  (3)  now  becomes 

...      dM  .  . 


-,  1 

J-=.0,  ,-  —  —  9j,f 

If  y^  and  yz  are  of  the  same  sign,  x  and  -^—  have  the  same 

GLJC 

sign.     But   according  to   observation  the   wires  attract  if   the 
currents  flow  in  the  same  direction,  therefore  x  is  negative  and 

dM  . 

-p-  is  negative. 

ax 

If  x  =  0,  we  have  a  single  wire  carrying  a  current  of  inten- 
sity y^  +  yz.  The  electrokinetic  energy  Ly*  -f-  My$z  +  £#22  be- 
comes L  (^  +  y2)2  or  Lyl  +  2  i^2  +  Lyl  so  that  the  value  of 
M  when  #  =  0  is  2  i.  Hence,  as  M  is  a  decreasing  function,  M 
is  less  than  2  i  always.  As  Jf  is  easily  seen  to  be  zero  when 
x  is  infinite,  it  must  be  positive  for  all  values  of  x. 

We  have,  then, 

T  =  Ai?  4-  Lyl  +  My&  +  Lyl  (4) 

where  A  and  L  are  positive  constants  and  M  is  a  positive  decreas- 
ing function  of  x  always  less  than  2  L.    L  is  called  the  coefficient 
of  self-induction  of  either  wire  per  unit  of  length,  and  M  the  coeffi- 
cient of  mutual  induction  of  the  pair  of  wires  per  unit  of  length. 
Our  Lagrangian  equations  are 


=  ^-^,  (6) 

=  ^2-^2.  (7) 

54.  Induced  Currents,  (a)  Suppose  that  no  current  is  flow- 
ing in  either  of  the  wires  considered  in  the  last  section,  and 
that  the  first  wire  is  suddenly  connected  with  a  battery  fur- 
nishing an  electromotive  force  JS^  and  that  thereby  a  current 


CHAP.  V]  INDUCED  CURRENTS  107 

y:  is   impulsively  established.     Then,   by  Thomson's  Theorem- 
(v.  Art.  30),  such  impulsive  velocities  must  be  set  up  in  the 
system  as  to  make  the  energy  have  the   least   possible   value 
consistent  with  the  velocity  caused  by  the  applied  impulse. 
If  the  current  yl  set  up  in  the  first  wire  has  the  intensity  i, 

Li2 


and  making  T  a  minimum,  we  have 


whence  x  =  0,  and  the  second  wire  will  have  no  initial  velocity, 

Mi 

and  y2  =  —  —  —  ,  and  a  current  will  be  set  up  impulsively  in  the 

aJj 

second  wire,  of  intensity  proportional  to  the  intensity  of  the  cur- 
rent in  the  first  wire.  Since,  as  we  have  seen,  M  and  L  are  both 
positive,  y2  is  negative,  and  this  so-called  induced  current  will  be 
opposite  in  direction  to  the  impressed  current  y^.  This  impulsively 
induced  current  is  soon  destroyed  by  the  resistance  of  the  wire. 
(6)  Suppose  that  a  steady  current  y^  caused  by  the  electro- 
motive force  .Ej,  is  flowing  in  the  first  wire  while  the  second 
wire  is  inert,,  and  that  E  ,  and  consequently  y^  is  impulsively 
destroyed  by  suddenly  disconnecting  the  battery.  This  amounts 
to  impulsively  applying  to  the  first  wire  the  additional  electro- 
motive force  —  E^  If  the  system  were  initially  inert,  this,  as 
we  have  just  seen  in  (a),  would  immediately  set  up  the  induced 

current  yz  =  —  —  in  the  second  wire,  and  we  should  have  y  =  —  i, 

Mi 

?/2  =  -  —  ,  as  the  immediate  result  of  our  impulsive  action.  Combine 

'—  1  < 

this  with  the  initial  motion  in  our  actual  problem,  yl  =  i,  yz  =  0, 

Mi 
and  we  get  for  our  actual  result  y1  =  0,  i/z  =  -  —  (v.  Art.  34). 

'—  lj 

So  that  if  our  first  wire  is  suddenly  disconnected  from  the 
battery,  an  induced  current  whose  direction  is  the  same  as  that 
of  the  original  current  is  set  up  in  the  second  wire.  It  is,  how- 
ever, soon  destroyed  by  the  resistance  of  the  wire. 


108  APPLICATION  TO  PHYSICS  [ART.  54 

(c)  Suppose  that  we  have  a  current  y^  in  our  fixed  wire,  caused 
by  a  battery  of  electromotive  force  E^  and  no  current  in  our 
second  wire,  and  that  the  second  wire  is  made  to  move  away 
from  the  first.  Equations  (6)  and  (7)  of  the  preceding  section 
give  us 


dt 

—  C2Ly  — Mi/^)x 

dx 

(4£2  —  Jf12)-^=  2LCE  —  Ry}—M(E  —  Ry} 
dt 

.dM 

—  (ViLy  — My  )x • 

2       dx 

When  we  are  starting  to  move  the  second  wire, 
and  we  have 


s*  T2      Ti,r2N      ,      Ttr-  - 

(4:  L2  —  M2)  -£i  =  Myx , 

*  dt         ^1    dx 

ZN  dya  .  dM 

—M*}-2i  =  —  ZLyx— — 
dt  dx 

As  we  have  seen,  L,  M,  4  L?  —  M2,  are  positive  and  — —  is  nega- 

vKv 

tive.  Hence  the  current  y^  will  decrease  in  intensity,  and  a 
current  yz  having  the  same  direction  as  y^  will  be  induced  in 
the  moving  wire. 

The  phenomena  of  induced  currents  which  we  have  just 
inferred  from  our  Lagrangian  equations  are  entirely  confirmed 
by  observation  and  experiment. 


APPENDIX  A 

SYLLABUS.     DYNAMICS  OF  A  RIGID  BODY 

1.  D'Alembert's  Principle.    In  a  moving  system  of  particles  the 
resultant  of  all  the  forces  external  and  internal  that  act  on  any 
particle  is  called  the  effective  force  on  that  particle.    Its  rectangular 
components  are  mx,  my,  and  mi. 

The  science  of  rigid  dynamics  is  based  on  jyAlembert's  prin- 
ciple: In  any  moving  system  the  actual  forces  impressed  and 
internal,  and  the  effective  forces  reversed  in  direction,  form  a  set 
of  forces  in  equilibrium,  and  if  the  system  is  a  single  rigid  body, 
the  internal  forces  are  a  set  separately  in  equilibrium  and  may 
be  disregarded. 

It  follows  from  this  principle  that  in  any  moving  system  the 
actual  forces  and  the  effective  forces  are  mechanically  equivalent. 
Hence, 

(«)  Swo;  =  2-X", 

(1}  2m  [_yx  -x])-]  =  ^  \_yX  -  xY]} 

(c)  °2m  [x8x  +  l/8>j  +  z%z~\  =  2  [A'&B  +  Y&j  +  Z&z]. 

These  equations  may  be  put  into  words  as  follows : 

(a)  The  sum  of  those  components  which  have  a  given  direction 
is  the  same  for  the  effective  forces  and  for  the  actual  forces. 

(/>)  The  sum  of  the  moments  about  any  fixed  line  is  the  same  for 
the  effective  forces  and  for  the  actual  forces. 

(c)  In  any  displacement  of  the  system,  actual  or  hypothetical,  the 
work  done  by  the  effective  forces  is  equal  to  the  work  done  by  the 
actual  forces. 

Equations  (a)  and  (J)  are  called  differential  equations  of  motion 
kfor  the  system. 

2.  px  =  "S,mvx  =•  ^mx  and  is  the  linear  momentum  of  the  system  in 
the  A"   direction  ;   h..  =  2?/i  [yvx  —  xvy~]  —  "2,m\_yx  —  xy~]  and  is  the 
moment  of  momentum,  about  the  axis  of  Z. 

109 


110  APPENDIX  A 

Equations  (a)  and  (b~)  of  §  1  may  be  written,  respectively, 


and  §  1,  (a),  and  §  1,  (£),  may  now  be  stated  as  follows  : 

(a)  In  a  moving  system  the  rate  of  change  of  the  linear  momentum 

in  any  given  direction  is  equal  to  the  sum  of  those  components  of 

the  actual  forces  which  have  the  direction  in  question. 

(&)  In  a  moving  system  the  rate  of  change  of  the  moment  of 

momentum  about  any  line  fixed  in  space  is  equal  to  the  sum  of  the 

moments  of  the  actual  forces  about  that  line. 

„  * 

3.  Center  of  Gravity.  x 


at 

dx 

Hence  px  =  zmx  =  M  —  ; 

dtj 

or,  the  linear  momentum  in  the  X  direction  is  what  it  would  be  if 
the  whole  system  were  concentrated  at  its  center  of  gravity. 
_  dx      _ 


or,  the  moment  of  momentum  about  the  axis  of  Z  is  what  it  would  be  if 
the  whole  mass  were  concentrated  at  the  center  of  gravity  plus  what 
it  would  be  if  the  center  of  gravity  were  at  rest  at  the  origin  and  the 
actual  motion  were  what  the  relative  motion  about  the  moving  center 
of  gravity  really  is. 

4.  The  motion  of  the  center  of  gravity  of  a  moving  system  is  the 
same  as  if  all  the  mass  were  concentrated  there  and  all  the  actual 
forces,  unchanged  in  direction  and  magnitude,  were  applied  there. 

The  motion  about  the  center  of  gravity  is  the  same  as  if  the  center 
of  gravity  were  fixed  in  space  and  the  actual  forces  were  unchanged 
in  magnitude,  direction,  and  point  of  application. 

5.  If  the  system  is  a  rigid  body  containing  a  fixed  axis, 


where  Mk®  =  M(h*  +  k*)  and  is  the  moment  of  inertia  about  the  axis, 
and  where  <o  is  the  angular  velocity  of  the  body. 

Equation  (&)  of  §  1  becomes  .I/A-'2  —  =  JV,  where  N  is  the  sum 
of  the  moments  of  the  impressed  forces  about  the  fixed  axis. 


APPENDIX  A 


111 


6.  If  the  system  is  a  rigid  body  containing  a  fixed  point  and  <ax, 
<ay,  w2,  are  its  angular  velocities  about  three  axes  fixed  in  space  and 
passing  through  the  fixed  point, 

hz  =  Cwz  —  E<ox  —  Dwv, 

where  C  is  the  moment  of  inertia  about  the  axis  of  Z,  and  D  and  E 
are  the  products  of  inertia  about  the  axes  of  X  and  F,  respectively ; 
that  is, 

C  =  2m  (ar2  +  T/2),         D  =  2myz,         E  = 

Equation  (It)  of  §  1  becomes 

dli_z  _  c  du>z  _  E  dwx  _  ^  dwv  _       _  E^ 

dt  dt  dt  dt 


where  A"  is  the  sum  of  the  moments  of  the  impressed  forces  about 
the  axis  of  Z. 

7.  Euler's  Equations.  If  the  system  is  a  rigid  body  containing 
a  fixed  point  and  coa,  w2,  o>3,  are  its  angular  velocities  about  the 
principal  axes  of  inertia  through  that  point  (a  set  of  axes  fixed  in 
the  body  and  moving  with  it),  equation  (&)  of  §  1  becomes 


8.  Euler's  Angles.  Euler's  angles  0,  \j/,  <£,  are  coordinates  of  a  mov- 
ing system  of  rectangular  axes  At  B,  C,  referred  to  a  fixed  system 
A',  F,  Z,  having  the  same 
origin  O.  0  is  the  colati- 
tude  and  $  the  longitude 
of  the  moving  axis  of  C 
in  the  fixed  system  (re- 
garded as  a  spherical  sys- 
tem with  the  fixed  axis  of 
Z  as  the  polar  axis),  and 
<f>  is  the  angle  made  by 
the  moving  C.I -plane  with 
the  plane  through  the  fixed 
axis  of  Z  and  the  moving 
axis  of  C. 


112  APPENDIX  A 

< 
We  have  /  0  U        KV  =  6  sin  <f>  —  \j/  sin  6  cos  <£, 

. 
f  o)0  =  0  cos  <i  4-  li  sin  6  sin  <&, 

<P       /  ft 

•       /»  i    ;  »     )TI  ,. 

w  =  w  cos  6*  +  o> ; 

v.      3  /, 

and  (to,..  =  —  0  sin  ^  -f-  <i  sin  0  cos  i^. 

Ic"  \\     c'j 

<  <„„  =  6  cos  i^  4-  i  sin  0  sin  ^, 

/V'ls    /  .         ,_J. 

a),  =  d>  COS  0  +  i/f. 

A  v 

9.  £  mv2,  the  kinetic  energy  of  a  particle,  becomes 

—  [x2  +  y2  +  «2]  in  rectangular  coordinates, 

L 

—  [r*  +  r-2^2]  in  polar  coordinates, 

A 

—  [r2  +  r2^  4-  sin204'2)]  in  spherical  coordinates. 

7/7 

10.   2,  "o" w2)  the  kinetic  energy  of  a  moving  system,  becomes 

^"^  ^ 

"V  —  (ic2  4-  y2  4-  z2)  in  rectangular  coordinates ; 

—  ka(a2  if  a  rigid  body  contains  a  fixed  axis ; 

M\/dx^      A^A2!  ,   M   „  2 

—  (  —  )  4-{-77J     +17  **?  "  the  body  is  tree  and  the  motion 
^  \_\dt /       \dt I  J       & 

is  two-dimensional ; 


-  2  -Fw^J    if   the 
body  is  rigid  and  contains  a  fixed  point ; 

%\_A<j>-f  +-Z?<i>22  +  Cfc>82]  if  the  body  is  rigid  and  the  axes  are 
the  principal  axes  for  the  fixed  point. 

11.  Impulsive  Forces.  In  a  system  acted  on  by  impulsive  forces, 
the  resultant  of  all  the  impulsive  forces  external  and  internal  that 
act  on  any  particle  is  called  the  effective  impulsive  force  on  that 
particle.  Its  rectangular  components  are  m  (x^  —  o;0),  m  (y^  —  y0), 
(mzl  —  2Q).  D'Alembert's  principle  holds  for  impulsive  forces.  It 


APPENDIX  A  113 

follows  that  in  any  system  acted  on  by  impulsive  forces,  the  actual 
forces  and  the  effective  forces  are  mechanically  equivalent.   Hence, 


(a) 

(6)   2m  [2/1x1  -  xli/l  -  y0x0  +  a^J  =  2  \yX  -  x  F], 

(c)    2m  [(i1  -  zfl)  &r  +  (^  -  $0)  By  +  (^  -  «0)  Sz] 


These  equations  may  be  put  into  words  as  follows  : 

(a)  The  sum  of  the  components  which  have  a  given  direction 
is  the  same  for  the  effective  impulsive  forces  and  for  the  actual 
impulsive  forces. 

(b)  The  sum  of  the  moments  about  a  given  line  is  the  same  for 
the  effective  impulsive  forces  and  for  the  actual  impulsive  forces. 

(c)  In  any  displacement  of  the  system,  actual  or  hypothetical, 
the  sum  of  the  virtual  moments  of  the  effective  impulsive  forces  is 
equal  to  the  sum  of  the  virtual  moments  of  the  actual  impulsive 
forces. 

Equations  («)  and  (&)  are  the  equations  for  the  initial  motion 
under  impulsive  forces,  (a)  and  (&)  may  be  restated  as  follows  : 

In  a  system  acted  on  by  impulsive  forces,  the  total  change  in  the 
linear  momentum  in  any  given  direction  is  equal  to  the  sum  of  those 
components  of  the  actual  impulsive  forces  which  have  the  direction 
in  question. 

In  a  system  acted  on  by  impulsive  forces,  the  total  change  in  the 
moment  of  momentum  about  any  fixed  line  is  equal  to  the  sum  of 
the  moments  of  the  actual  impulsive  forces  about  that  line. 

Section  4  holds  unaltered  for  impulsive  forces. 


APPENDIX  B 


THE  CALCULUS  OF  VARIATIOXS 

1.  The  calculus  of  variations  owed  its  origin  to  the  attempt  to 
solve  a  very  interesting  class  of  problems  in  maxima  and  minima 
in  which  it  is  required  to  find  the  form  of  a  function  such  that  the 
definite  integral  of  an  expression  involving  that  function  and  its 
derivatives  shall  be  a  maximum  or  a  minimum. 

Take  a  simple  case:  If  y=f(x),  let  it  be  required  to  deter- 
mine the  form  of  the  function  /,  so  that  J  <f>\x,  y,  -j-  \dx  shall  be 
a  maximum  or  a  minimum.  **** 

Let  /(a;)  and  F(x)  be  two 
possible  forms  of  the  function. 
Consider  their  graphs  y  =/(#) 
and  y  —  F(x). 

If  ,(*)is  F(x)-f(x),y(x) 
can  be  regarded  as  the  incre- 
ment given  to  y  by  changing 
the  form  of  the  function  from 
f(x)  to  F(x~),  the  value  of  the 
independent  variable  x  being 
held  fast. 

This  increment  of  y,rj(x),  is  called  the  variation  of  y  and  is  written 
8y ;  it  is  a  function  of  x}  and  usually  a  wholly  arbitrary  function  of  ./•. 

The  corresponding  increment  in  y',  where  y'  =  -j-  >  can  be  shown  to  be 

dii 
i/(#),  and  is  the  variation  of  ?/',  and  is  written  8y',  or  8  -j-  •    Obviously, 

ClX 


A. 

dx 


(1) 


If  an  infinitesimal  increment  8y  is  given  to  y,  it  is  proved  in  the 

differential  calculus  that  —  <f>(y~)8y  differs  by  an  infinitesimal  of 

ay 

higher  order  than  £?/  from  the  increment  produced  in  <£(//).    This 

114 


APPENDIX  B  115 

approximate  increment  is  called  the  variation  of  <f>  (y),  so  that 

(2) 


Similarly,  ^  (y,  y  ')  =       3y  +       8y',  (3) 

or  since  x  is  not  varied, 


As  d$(y}  =  — 

r\    .  Q  _I 

and  cty  (y,  ?/')  =  ^  efy  +  ^  <fy', 

^y  cy 

we  can  calculate  variations  by  the  familiar  formulas  and  processes 
used  in  calculating  differentials. 

2.  Let  a  be  an  independent  parameter.  Then  y  =  f(x)  +  at]  (a?), 
or  y  =.f(x)  +  tfSy,  is  any  one  of  a  family  of  curves  including 
y  =f(x)  (corresponding  to  a  =  0)  and  y  =f(x)  +  rj  (#)  (corresponding 
to  a  =5=  1).  If  XQ  and  cCj  are  fixed  values,  and  if 


I(a)=  \      <f>(x,y  +  <%,  y' 

JxQ 

I  (a)  is  a  function  of  the  parameter  a  only.  A  necessary  condition 
that  I  (a),  a  function  of  a  single  variable  a,  should  be  a  maximum 
or  a  minimum  when  a  =  0  is  /'(a)  =  0  when  a  =  0. 


when  a  =  0. 

That  is,  ^'(°)=  f  '^dx.  (v,  §  1,  (4)) 

A  necessary  condition,  then,  that   I     <f>(x,  y,  y'^dx  should  be  a 

A)  /•»! 

maximum  or  a  minimum  when  ?/  =  f(x}  is    I     I 

J* 


116 


APPENDIX  B 


I  &<f)dx  is  taken  as  the  definition  of  the  variation  of  I  <j>dx, 
our  necessary  condition  is  usually  written 

8  f  l<i>(x,y,y')dx  =  Q. 

J* 


and 


How  this  condition  helps  toward  the  determination  of  the  form 
of  f(x)  can  be  seen  from  an  example. 

3.  Let  it  be  required  to  find  the  form  of  the  shortest  curve 
y  =/(#)  joining  two  given  points  (XQ,  y0)  and  (x^  yj.   Here,  since 


ds  =  Vl  +  i/^dx, 
1=  f  l Vl  +  y»dxt 

J  x* 


and  /  is  to  be  made  a  minimum. 

S/  =  f   8  Vl  +  //«  .  dx 

^^dx 


dx 


r*i y^ 

J*.  ^+ 


Integrating  by  parts,  this  last  reduces  to 


=  0, 

since,  as  the  ends  of  the  path  are  given,  Sy  =  0  when  x  =  x0  and 
when  x  =  xv   Then  87  =  0  if 

C    ~-jJL=tydx  =  0; 

o  «/ 

but  since  our  8y  (that  is,  i/(«)),  is  a  function  which  is  wholly  arbitrary, 


APPENDIX  B  117 


d         y' 

"• 


the  other  factor,  —  --  .  "•         ,  must  be  equal  to  zero  if  the  integral 

.  '  ' 

is  to  vanish. 


dx         + 

' 


This  gives  us  —  .  =  C. 


y  =c. 

y  =  ex  +  d ; 
and  the  required  curve  is  a  straight  line. 

4.  In  our  more  general  problem  it  may  be  shown  in  like  manner  that 

-8  f  V(*,j 


leads  to  a  differential  equation  between  y  and  x  and  so  determines  y 
as  a  function  of  x. 

Of  course  8/  =  0  is  not  a  sufficient  condition  for  the  existence  of 
either  a  maximum  or  a  minimum  and  does  not  enable  us  to  discrim- 
inate between  maxima  and  minima,  but  like  the  necessary  condition 

—  =  0  for  a  maximum  or  minimum  value  in  a  function  of  a  single 
variable,  it  often  is  enough  to  lead  us  to  the  solution  of  the  problem. 

5.  Let  us  now  generalize  a  little.    Let  x,  y,  z,  •  •  -,  be  functions  of 

dx  dy  dz 

an  independent  variable  r,  and  let  cc  =  —  >  y'  =  -r->  «'  =  —>..:. 

dr  dr  dr 

Suppose  we  have  a  function  </>(/•,  x,  y,  z,  •  •  •,  x',  y1,  «',  •  •  •). 

By  changing  the  forms  of  the  functions  but  holding  r  fast,  let 
x,  y,  z, '  •  •,  be  given  the  increments  £(?*),  ij(r),  £(V), 

x  then  becomes  x  +  £  (r),  y  becomes  y  +  rj  (r),  z  becomes  z-\-  £  (/•),•••; 
x'  becomes  x'  +  £'(r)>  y'  becomes  y'  + if (r),  z1  becomes  z'  +  £'(r)>  •  •  •• 

£(r),  £'(/•),  are  the  variations  of  x  and  x'  and  are  written  £e 
and  £c'.  Obviously,  ^ 


The  increment  produced  in  <£  when  infinitesimal  increments  &r, 
8y,  •  •  -,  8x',  &y',  •  •  .,  are  given  to  the  dependent  variables  and  to  their 
derivatives  with  respect  to  r  is  known  to  differ  from 


by  terms  of  higher  order  than  the  variations  involved.    This  approxi- 
mate increment  is  called  the  variation  of  <f>  and  is  written  8<£.    It  is 


118  APPENDIX  B 

found  in  any  case  precisely  as  d<f>,  the  complete  differential  of  <j>, 
is  found. 


rri 

That   I      8^5)  (r,  x,  y,  •  •  •  ,  x',  y',  •  •  •  }dr  =  0  is  a  necessary  condition 

\J  Tf. 

rri 

that    I      <£(r,  x,  y,  •  •  •  ,  x',  y',  •  -  •  )  dr  should  be  a  maximum  or  a 

\J  T 

minimum  can  be  established  by  the  reasoning  used  in  the  case  of 
rxi  rri 

I      <j>(%,  y,  y')dx.    The  integral  I      8<f>dr  is  called  the  variation 

Jx0  Jr0 

rri  rri  rri 

of  I      (f)  dr,  so  that  81      $  dr  =  I      8<f>  dr. 

Jr0  JrQ  Jr<i 

6.  It  should  be  noted  that  our  important  formulas 

frdx       d 
dr       dr 

rri  rr! 

8  I      <f>dr  =   I 

Jr.  Jr.. 


and 

hold  only  when  r  is  the  independent  variable  which  is  held  fast 
when  the  forms  of  the  functions  are  varied;  that  is,  when  8r  is 

supposed  to  be  zero. 

dij 

If  x  and  y  are  functions  of  r  and  we  need  8-f^>  we  get  it  in- 
directly thus :  d          , 

dx      x1 

«dy     «yf        y' «       1  *       i[d  .    .  dy  d  ,~\ 

8-^  =  8—.=—^-i:28xl  +  —  8u'=:  —,\-r8y  —  -f- —  8x  , 
dx         x  x  x    '         x\_dr  dx  dr     J 

-dy        d  dy  d  . 

so  that  8  -r-  =  —  oy  —  -r-  —  8x.  (1) 

dx      dx  dx  dx 

If  we  need  8  I  <f>dx,  we  must  change  our  variable  of  integration 
to  r.  J 


C     dr 
<f>dx  =  S  I  <t>j;dr. 


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